University Linear Algebra — Vector Spaces and Subspaces Worksheet

All 11 questions

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

1. Q1Vectors in n-Space and Linear Combinations

   In ${ℝ}^5$, let $\overrightarrow{u}=(1,-2,0,3,1)$, $\overrightarrow{v}=(2,1,-1,0,4)$ and $\overrightarrow{w}=(0,3,2,-1,-2)$.

   1. Compute the linear combination $2\overrightarrow{u}-3\overrightarrow{v}+\overrightarrow{w}$.
   2. Solve $3(\overrightarrow{x}-\overrightarrow{u})=\overrightarrow{x}+2\overrightarrow{v}$ for the vector $\overrightarrow{x}\in {ℝ}^5$.
   3. Is $\overrightarrow{v}$ a scalar multiple of $\overrightarrow{u}$? Justify in one line.
2. Q2Vectors in n-Space and Linear Combinations

   Consider the linear system $$\begin{gathered} x_1\phantom{+x_2}+2x_3\&=7 \\ 2x_1+x_2\phantom{+2x_3}\&=5 \\ 3x_2+x_3\&=-1 \\ -x_1+x_2+x_3\&=-2 \end{gathered}$$

   1. Rewrite it as a single vector equation $x_1{\overrightarrow{a}}_1+x_2{\overrightarrow{a}}_2+x_3{\overrightarrow{a}}_3=\overrightarrow{b}$ in ${ℝ}^4$, naming the four vectors.
   2. Find the weights $x_1,x_2,x_3$, and check them by computing the linear combination.
   3. Without solving anything further, write $(14,10,-2,-4)$ as a linear combination of ${\overrightarrow{a}}_1,{\overrightarrow{a}}_2,{\overrightarrow{a}}_3$.
3. Q3The Vector Space Axioms

   On $P_2$, the polynomials of degree at most $2$, keep the usual addition of polynomials but define a new scalar multiplication that discards the $x^2$ term: $$c\odot (a_0+a_{1}x+a_2{x}^2)=c{a}_0+c{a}_{1}x.$$ The addition axioms all hold, since addition is unchanged. Check the four scalar-multiplication axioms, $$c\odot (p+q)=c\odot p+c\odot q,(c+d)\odot p=c\odot p+d\odot p,c\odot (d\odot p)=(cd)\odot p,1\odot p=p,$$ and show that exactly one of them fails, with a specific counterexample.

4. Q4The Vector Space Axioms

   On ${ℝ}^2$, keep the usual scalar multiplication $c(x,y)=(cx,cy)$ but define a new addition $$(x_1,y_1)\oplus (x_2,y_2)=(x_1+x_2+1,y_1+y_2-2).$$

   1. Find the vector $\overrightarrow{z}$ with $\overrightarrow{u}\oplus \overrightarrow{z}=\overrightarrow{u}$ for every $\overrightarrow{u}$ (the zero vector for $\oplus$).
   2. Find the negative of $(3,-2)$ for $\oplus$, that is, the vector $\overrightarrow{n}$ with $(3,-2)\oplus \overrightarrow{n}=\overrightarrow{z}$.
   3. Show that the axiom $(c+d)\overrightarrow{u}=c\overrightarrow{u}\oplus d\overrightarrow{u}$ fails, with a specific counterexample.
5. Q5Consequences of the Axioms

   Here is a proof that $0\overrightarrow{v}=\overrightarrow{0}$ for every vector $\overrightarrow{v}$ in any vector space $V$. Here $0$ is the real number and $\overrightarrow{0}$ the zero vector of $V$. For each numbered step, give its justification in words: the property of real numbers, or the vector space axiom, that it uses. $$\begin{gathered} 0\overrightarrow{v}\&=(0+0)\overrightarrow{v}\&\&\text{(1)} \\ \&=0\overrightarrow{v}+0\overrightarrow{v}\&\&\text{(2)} \end{gathered}$$ Add the vector $-(0\overrightarrow{v})$ to both sides of $0\overrightarrow{v}=0\overrightarrow{v}+0\overrightarrow{v}$: $$\begin{gathered} 0\overrightarrow{v}+(-(0\overrightarrow{v}))\&=(0\overrightarrow{v}+0\overrightarrow{v})+(-(0\overrightarrow{v}))\&\&\text{(3)} \\ \overrightarrow{0}\&=0\overrightarrow{v}+(0\overrightarrow{v}+(-(0\overrightarrow{v})\left)\right)\&\&\text{(4)} \\ \overrightarrow{0}\&=0\overrightarrow{v}+\overrightarrow{0}\&\&\text{(5)} \\ \overrightarrow{0}\&=0\overrightarrow{v}\&\&\text{(6)} \end{gathered}$$

6. Q6Spaces of Polynomials, Matrices and Functions

   Operations in three spaces that are not ${ℝ}^n$, each with its usual addition and scalar multiplication.

   1. In $P_3$, let $p(x)=2-x+3x^3$ and $q(x)=1+4x-x^2+x^3$. Compute $3p-2q$.
   2. In $M_{2\times 3}$, let $A=\left[\begin{array}{ccc}1& 0& -2\\ 3& -1& 4\end{array}\right]$ and $B=\left[\begin{array}{ccc}0& 5& 1\\ -2& 2& 3\end{array}\right]$. Compute $2A-B$, and write down the zero vector of $M_{2\times 3}$ and the negative of $A$.
   3. In the space $F(ℝ)$ of all functions $ℝ\to ℝ$, let $f(x)=e^x$ and $g(x)=x^2$. Write down the function $3f-2g$ and its value at $x=0$. What is the zero vector of $F(ℝ)$?
7. Q7Spaces of Polynomials, Matrices and Functions

   In $M_{2\times 2}$, let $$E=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right],F=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],G=\left[\begin{array}{cc}1& 1\\ 0& -1\end{array}\right],H=\left[\begin{array}{cc}3& -2\\ -1& 1\end{array}\right].$$

   1. Find real numbers $a,b,c$ with $aE+bF+cG=H$, writing down the four equations that the entries give.
   2. Explain in one sentence why this is the same computation as writing a vector of ${ℝ}^4$ as a linear combination of three others.
8. Q8The Subspace Test

   Let $W=\{p\in P_3\mid p(1)=p(-1)\}$.

   1. Use the subspace test to show that $W$ is a subspace of $P_3$.
   2. Writing $p(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3$, turn the condition $p(1)=p(-1)$ into a condition on the coefficients, and decide whether $4-3x+x^2+3x^3$ and $1+x+x^3$ lie in $W$.
9. Q9The Subspace Test

   Each set below carries the usual operations. Decide whether it is a subspace. If it is, verify the three conditions; if it is not, name one condition that fails and give a specific counterexample.

   1. $W_1=\{\overrightarrow{x}\in {ℝ}^4\mid x_1+x_2+x_3+x_4=1\}$
   2. $W_2=\{\overrightarrow{x}\in {ℝ}^4\mid x_1{x}_4=0\}$
   3. $W_3=\{A\in M_{2\times 2}\mid A^T=A\}$
   4. $W_4=\{\overrightarrow{x}\in {ℝ}^4\mid x_2\ge |x_1|\}$
10. Q10Solution Sets of Homogeneous Systems as Subspaces

    Let $$A=\left[\begin{array}{cccc}1& 2& 0& -1\\ 2& 4& 1& 1\\ -1& -2& 1& 4\end{array}\right],W=\{\overrightarrow{x}\in {ℝ}^4\mid A\overrightarrow{x}=\overrightarrow{0}\}.$$

    1. Verify that $\overrightarrow{u}=(-2,1,0,0)$ and $\overrightarrow{v}=(1,0,-3,1)$ lie in $W$.
    2. Without solving the system, explain why $3\overrightarrow{u}-2\overrightarrow{v}$ lies in $W$, and write it down.
    3. The system $A\overrightarrow{x}=(1,2,-1)$ has the solution $(1,0,0,0)$. Is its solution set a subspace of ${ℝ}^4$? Justify.
11. Q11Synthesis — drawing on several topics in this unit

    Let $W=\{p\in P_3\mid p(1)=0\text{and}{p}^{\prime }(1)=0\}$, where $p^{\prime }$ is the derivative of $p$.

    1. Use the subspace test to show that $W$ is a subspace of $P_3$.
    2. Writing $p(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3$, turn the two conditions into a homogeneous linear system in $a_0,a_1,a_2,a_3$ and solve it.
    3. Deduce that every $p\in W$ is a linear combination of $q_1(x)=1-2x+x^2$ and $q_2(x)=2-3x+x^3$.
    4. Decide whether $7-12x+3x^2+2x^3$ and $x^3-1$ lie in $W$; for any that does, give its weights on $q_1$ and $q_2$.

The 8 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

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 Vector Spaces and Subspaces notes cover  About the Vector Spaces and Subspaces 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.

Vectors in n-space, and a system as one vector equation

Q1 is arithmetic in five coordinates, and its point is that nothing new happens there: a linear combination is computed one coordinate at a time, and an equation between vectors is solved with the same algebra as an equation between numbers — collect the unknown vector on one side, divide by its coefficient. Part (c) asks for a one-line justification, and the cleanest one compares a single pair of coordinates that no scalar can match.

Q2 is the reading that the whole course is built on. A system of four equations in three unknowns is the same statement as "b can be built from the three columns, and the unknowns are the weights".

The axioms, tested on operations that are not the usual ones

A vector space is a set with an addition and a scalar multiplication satisfying a fixed list of axioms. Q3 and Q4 each change one operation and ask which axioms survive. The method is the same in both: write each axiom out for the new operation, compute both sides separately, and compare.

Q4 also asks for the zero vector and a negative under the new addition. Neither is necessarily the one you expect. Find the zero vector from its defining property — the vector that leaves every u unchanged — by solving for its coordinates, and find the negative the same way, as the vector that brings (3, −2) to that zero.

A proof from the axioms

Q5 gives a complete proof that zero times any vector is the zero vector, and asks only for the reason behind each numbered step. It is the model for every short proof in the course: each line is licensed by exactly one axiom or one property of real numbers, and naming it is the whole job.

Polynomials, matrices and functions are vectors too

Q6 does the same arithmetic in three spaces that are not n-space: polynomials of degree at most 3, 2 × 3 matrices, and all functions from the reals to the reals. The questions about the zero vector are the ones to pause on — the zero of a matrix space is a matrix, and the zero of a function space is a function, not the number 0.

Q7 sets up a linear combination of four 2 × 2 matrices and asks for the weights. Matching the four entries gives four equations in three unknowns. Part (b) is the idea behind coordinates, which the course develops two sets later: a 2 × 2 matrix is four numbers in a fixed order, so the question is a linear-combination question in four coordinates in disguise.

The subspace test

A subspace is a subset that is a vector space in its own right under the operations it inherits. You do not re-check every axiom; three conditions are enough.

Q8 applies the test to the polynomials that take the same value at 1 and at −1. The verification must be general: take two arbitrary members and an arbitrary scalar, and show the defining condition still holds. Part (b) translates the condition into a statement about the coefficients — substitute 1 and −1 into the general cubic and see which coefficients survive the comparison — and then uses it to classify two specific polynomials.

Q9 gives four sets and asks you to decide. This is where the decision matters more than the algebra, and there are recognisable signs.

The homogeneous solution set is a subspace

Q10 connects the subspace test to systems. Part (a) is a substitution check. Part (b) is the reason homogeneous systems matter in this course: because A(su + tv) = sAu + tAv, any linear combination of solutions of Ax = 0 is again a solution — no solving needed. Part (c) asks the same question for a system with a non-zero right-hand side; settle it with the subspace test, starting from the condition that is quickest to check.

The synthesis question

Q11 follows one subset of the cubic polynomials through the whole set. Part (a) is Q8's subspace test with two conditions instead of one — the derivative condition behaves under addition and scaling because differentiation does. Part (b) is Q8(b) extended: turn both conditions into equations in the four coefficients and solve the homogeneous system, leaving the free coefficients as parameters. Part (c) reads that parametric solution as a linear combination, as in Q2 and Q10. Part (d) tests membership against the conditions and, where a polynomial is in the subspace, reads its weights off the free coefficients.

Preview all 7 pages

Click any page to open the full PDF.

Getting the most out of it

Write the definition down before you test it

Every question in this set is decided by a definition — of a linear combination, of an axiom, of a subspace. Write it out in the notation of the question before you compute, and most of the work is choosing what to substitute.

Prove "always" in general, disprove it with one example

To show a condition holds, use arbitrary vectors and scalars. To show it fails, pick the simplest specific ones you can — small integers, a negative scalar — and write the numbers out. Mixing the two is the most common way to lose marks on a subspace question.

Check the zero vector first

It is the fastest of the three conditions, and it rules out a whole family of sets — anything defined by an equation with a non-zero constant, and the solution set of any system that is not homogeneous.

Translate to coefficients

A condition on a polynomial or a matrix becomes a condition on its coefficients or entries. Once it is written that way, it is a linear system, and the tools from CEGEP apply.

Common questions