University Linear Algebra Basis, Dimension and Coordinates Free · no sign-up

University Linear Algebra — Basis, Dimension and Coordinates Worksheet

By Marius — private math tutor in Montreal. Updated 19 September 2026

The set where a vector space gets a size and a coordinate system. Deciding whether a set is a basis, and proving it from the definition when the space is not n-space; finding a basis of a subspace described by equations, and reading off its dimension; the counting arguments that answer a question without solving a system; completing an independent set to a basis; and coordinate vectors, which turn any finite-dimensional space into n-space and back. Have a look on this page, then print the free PDF when you want to write on it.

Page 1 of the University Linear Algebra Basis, Dimension and Coordinates practice worksheet

Practice worksheet — free PDF

6 pages 9 questions Letter size, print-ready

Download the worksheet Read the questions here

No email, no account, no watermark. Teachers: photocopy it for your classes freely. Worked solutions and 6 harder problems come with the University Linear Algebra bundle.

All 9 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

Q1Bases and the Standard Bases

Decide whether each set is a basis of $ℝ^{4}$ . Where it is not, write down an explicit dependence relation among its vectors.
1. ${\vec{v}}_{1} = (1, 0, 2, 1)$ , ${\vec{v}}_{2} = (0, 1, 1, - 1)$ , ${\vec{v}}_{3} = (1, 1, 0, 2)$ , ${\vec{v}}_{4} = (2, 1, 3, 1)$
2. ${\vec{w}}_{1} = (1, - 1, 2, 0)$ , ${\vec{w}}_{2} = (0, 1, 1, 3)$ , ${\vec{w}}_{3} = (2, - 3, 3, - 3)$ , ${\vec{w}}_{4} = (1, 1, 0, 1)$
Q2Bases and the Standard Bases

Write down the standard basis of $M_{2 \times 2}$ . Then show, directly from the definition of a basis, that $ℬ = {[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}], [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]}$ is also a basis of $M_{2 \times 2}$ .
Q3The Dimension of a Vector Space

Find a basis and the dimension of each subspace.
1. $W = {(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) \in ℝ^{5} ∣ x_{3} = x_{1} + x_{2} and x_{4} = 2 x_{5}}$
2. $U = {A \in M_{2 \times 3} ∣ each row of A has entries summing to 0}$
Q4The Dimension of a Vector Space

Answer each part without solving any system.
1. State $\dim P_{4}$ and $\dim M_{3 \times 2}$ .
2. Explain why any six polynomials in $P_{4}$ are linearly dependent.
3. Can five matrices span $M_{3 \times 2}$ ? Explain.
4. Someone has checked that five particular polynomials in $P_{4}$ are linearly independent. Explain why they must form a basis of $P_{4}$ .
Q5Reducing a Spanning Set and Extending an Independent Set

The polynomials $p_{1} (x) = 1 + x^{2}$ and $p_{2} (x) = x - x^{3}$ are linearly independent in $P_{3}$ . Extend ${p_{1}, p_{2}}$ to a basis of $P_{3}$ by adding polynomials from the standard basis ${1, x, x^{2}, x^{3}}$ , and justify that the result is a basis.
Q6Coordinate Vectors Relative to a Basis

The vectors ${\vec{b}}_{1} = (1, 1, 0, 0)$ , ${\vec{b}}_{2} = (0, 1, 1, 0)$ , ${\vec{b}}_{3} = (0, 0, 1, 1)$ , ${\vec{b}}_{4} = (0, 0, 0, 1)$ form a basis $ℬ$ of $ℝ^{4}$ . Find the coordinate vector $[\vec{x}]_{ℬ}$ of $\vec{x} = (3, 1, 4, 2)$ .
Q7Coordinate Vectors Relative to a Basis

The vectors ${\vec{b}}_{1} = (1, 0, 1, 0)$ , ${\vec{b}}_{2} = (0, 1, 0, 1)$ , ${\vec{b}}_{3} = (1, - 1, 0, 0)$ , ${\vec{b}}_{4} = (0, 0, 1, 1)$ form a basis $ℬ$ of $ℝ^{4}$ .
1. Find $\vec{v}$ if $[\vec{v}]_{ℬ} = (2, - 1, 3, 1)$ .
2. A vector $\vec{u}$ has $[\vec{u}]_{ℬ} = (1, 0, - 2, 4)$ . Find $[3 \vec{v} - 2 \vec{u}]_{ℬ}$ without finding $\vec{u}$ or solving any system.
Q8Coordinates in Polynomial and Matrix Spaces

Find each coordinate vector.
1. $[p]_{ℬ}$ , where $p (x) = 2 x^{2} - 5 x + 4$ and $ℬ = {1, x - 3, (x - 3)^{2}}$ is a basis of $P_{2}$ .
2. $[A]_{𝒞}$ , where $A = [\begin{matrix} 4 & - 1 \\ 3 & 2 \end{matrix}]$ and $𝒞 = {[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 1 & 1 \\ 0 & 0 \end{matrix}], [\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}], [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]}$ is a basis of $M_{2 \times 2}$ .
Q9Synthesis — drawing on several topics in this unit

Let $W = {p \in P_{3} ∣ p (0) = p (2)}$ , a subspace of $P_{3}$ , and let $ℬ = {1, x^{2} - 2 x, x^{3} - 4 x}$ .
1. Show that $ℬ$ is a basis of $W$ , and state $\dim W$ .
2. Show that $q (x) = x^{3} - 3 x^{2} + 2 x + 5$ lies in $W$ , and find $[q]_{ℬ}$ .

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

What does this set assume? This set stands on CEGEP Linear Algebra — row reduction, the inverse and the determinant are used without comment — and on the two sets before it: vector spaces and subspaces, and span and linear independence, in n-space and in the spaces of polynomials and matrices. A basis is exactly a set that spans and is independent, so every basis question here is those two questions asked together. Scope is bases and the standard bases, dimension, reducing a spanning set and extending an independent set, and coordinate vectors relative to a basis. The matrix that converts coordinates from one basis to another waits until after linear transformations, in the Change of Basis set. All spaces here are finite-dimensional and the scalars are real.

Which course is this for? In the public course calendars of Montreal universities, this material is part of the courses numbered MATH 133, MATH 204, MATH 251, MAST 234, MAT1250, MAT1260 and MAT472. Each course orders and weights the topics its own way, so check your own outline for what your exam covers. Which sets match your course.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the University Linear Algebra bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Basis, Dimension and Coordinates notes cover About the Basis, Dimension and Coordinates unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Is this set a basis?

A basis of a space is a set that spans it and is linearly independent. Q1 gives two sets of four vectors in four coordinates. With exactly as many vectors as the dimension, one test is enough: put the vectors in as columns of a square matrix and decide whether it is invertible — by row reduction or by the determinant.

n vectors in Rⁿ: basis ⇔ independent ⇔ spanning ⇔ the square matrix is invertible

"Not a basis" needs a relation. The question asks for an explicit dependence relation wherever a set fails. A zero determinant proves dependence but gives no weights, so for that case row reduce, take a value for the free variable, and write the relation out with its weights.

Q2 asks first for the standard basis of the 2 × 2 matrices — the four matrices with a single 1 and zeros elsewhere — and then for a proof, from the definition, that a different set of four matrices is also a basis. "From the definition" means both halves: show every 2 × 2 matrix is a combination of the four (solve for the weights in terms of the entries a, b, c, d) and show the only combination giving the zero matrix is the trivial one.

One system does both jobs. Solving the general system for arbitrary entries shows spanning; the same solution with every entry set to zero shows the weights must all be zero. Say explicitly that you have used it twice.

Dimension, and a basis for a subspace

Q3 describes two subspaces by conditions and asks for a basis and the dimension of each. The method is the one from the homogeneous systems of the first set: treat the conditions as equations, solve them, write the general member with its free parameters, and split it into one vector per parameter.

A basis from a description

Free parameters become basis vectors.

1
Solve the conditions
Express the dependent coordinates (or entries) in terms of the free ones.
2
Write the general member
Substitute, so the vector or matrix contains only the free parameters.
3
Split by parameter
Write it as parameter times vector, summed over the parameters. Those vectors span.
4
Check independence and count
Each vector has a 1 in its own parameter's position where the others have 0, so they are independent. The dimension is the number of parameters.

Q4 is answered without solving anything, and every part uses the same idea: once you know the dimension n of a space, the size of a set tells you a great deal about it.

What the dimension n guarantees. More than n vectors must be dependent. Fewer than n vectors cannot span. Exactly n vectors that are independent automatically span, and exactly n that span are automatically independent. Start part (a) from the standard basis of each space — count its members — and let the other three parts quote the right fact.

Extending an independent set

Q5 gives two independent cubic polynomials and asks you to complete them to a basis using members of the standard basis. The space has dimension four, so you need two more, and the question is which two.

Add one at a time, and test. Put the two given polynomials first and the four standard ones after them, as columns of coefficients, and row reduce. The pivot columns are an independent spanning set that keeps the given two. Alternatively, add a candidate and check that it is not a combination of what you already have. Either way, finish by saying why the four you have are a basis — four independent vectors in a four-dimensional space.

Coordinate vectors

Every vector is a combination of a basis in exactly one way, and the list of weights is its coordinate vector relative to that basis. Q6 asks for one: set up the combination, match the coordinates, and solve. The basis in Q6 is built so the system solves from the top down, one unknown per line.

[x]_B = (c₁, …, cₙ) means x = c₁b₁ + ⋯ + cₙbₙ

Q7 runs the definition the other way. Part (a) gives the coordinates and asks for the vector: no system at all, just compute the combination. Part (b) is the property that makes coordinates useful.

Coordinates are linear. The coordinate vector of a combination is the same combination of the coordinate vectors. So in Q7(b) you never need u itself: combine the two coordinate vectors you are given, with the same weights.

Coordinates in polynomial and matrix spaces

Q8 asks for coordinates relative to bases that are not standard. In part (a) the basis is built from powers of x − 3. Either expand each basis polynomial and match coefficients of 1, x and x², or rewrite p in powers of x − 3 directly by substituting x = (x − 3) + 3. In part (b) each basis matrix adds one more 1 to the previous one, so matching entries gives a system that solves from the last entry backwards.

The order of the basis matters. A coordinate vector lists weights in the order the basis is written. Reorder the basis and the coordinates reorder with it, so keep the order you were given.

The synthesis question

Q9 does the whole set on one subspace of the cubic polynomials. Part (a) needs two things: that each polynomial in the proposed basis lies in the subspace and that together they are a basis of it. Independence is a quick check here — look at the highest power in each. For spanning, find the dimension of the subspace independently, as in Q3, by turning the condition into an equation on the coefficients; then Q4's counting fact finishes the argument. Part (b) checks membership by substitution and then finds coordinates as in Q8, matching coefficients from the highest power down.

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Getting the most out of it

Count before you compute

Before any basis question, write down the dimension of the space and the number of vectors you have. Half the time the count settles the question, and the rest of the time it tells you which of spanning and independence you still need to check.

Know the standard bases by heart

Rⁿ, the polynomials of degree at most n, and the m × n matrices each have a standard basis you should be able to write down instantly. Their sizes are the dimensions you count with.

Turn everything into coordinates

A question about polynomials or matrices becomes a question about columns of numbers once you write each object as its coefficients or entries in a fixed order. Do that first, and the CEGEP row-reduction toolkit applies unchanged.

Check coordinates by rebuilding the vector

A coordinate vector is correct exactly when the combination it describes gives back the original vector. That check takes one line — always do it.

Want the solutions, or something more challenging?

The worksheet, the questions above and every explanation on this page stay free permanently. Three more PDFs exist for this topic — the worked answer key, a harder problem set, and the answer key to that. They come with the University Linear Algebra Solutions Bundle, beside the unit notes and the unit test, which is what keeps the rest of the series free.

What else exists for Basis, Dimension and Coordinates

Three PDFs · 12 pages · all three are in the bundle below.

Answer key — 3 pages. All 9 questions worked step by step, including the restrictions and the justifications. Not a list of final answers.
Challenge problems — 6 pages, 6 problems. A separate sheet at exam-plus difficulty covering the same 5 concepts. Harder than anything on the free sheet.
Challenge answer key — 3 pages. Every challenge problem worked to the same standard, with the checks shown.
PDF, letter size, print-ready.

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Common questions

Is this worksheet really free?

Yes — the questions are on this page to read, and the PDF downloads directly, no email and no account. The one paid item is optional: the complete University Linear Algebra Solutions Bundle, which covers every set at this level.

Which university courses is this for?

The course codes listed on this page are taken from the public course calendars of universities that teach linear algebra after CEGEP. Each course orders and weights the topics its own way — some teach coordinates together with bases, others postpone them to the change-of-basis material — so check the outline for your own section to see where this set falls in your term.

What do I need to know before starting this set?

CEGEP Linear Algebra, and the two previous sets of this course: Vector Spaces and Subspaces, and Span and Linear Independence.

Is the basis of a space unique?

No — a space other than the zero space has infinitely many bases. What is unique is their size: every basis of a finite-dimensional space has the same number of vectors, and that number is the dimension.

If I have the right number of vectors, do I need to check both spanning and independence?

No. When the number of vectors equals the dimension of the space, either property implies the other, so one check is enough. You do need to know the dimension first, which is why counting comes before computing.

What is the dimension of the zero space?

Zero. Its basis is the empty set, which spans the zero space and is, vacuously, independent.

Where is the change of basis matrix?

In the Change of Basis and Matrix Representations set, after linear transformations. This set finds coordinates relative to one basis at a time; converting between two bases with a matrix comes later, where it is presented together with the matrix of a transformation.

Can teachers use this in class?

Yes. Print and photocopy it for your own classes freely — I just ask that the tutorinmontreal.ca footer stays on the page.

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Yes — through one-on-one tutoring, in Montreal or online. Get in touch to arrange a session, or see the current rates.

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