University Linear Algebra — Span and Linear Independence Worksheet

All 10 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. Q1The Span of a Set of Vectors

   In ${ℝ}^4$ let $W=span\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2\}$, where ${\overrightarrow{v}}_1=(1,0,2,-1)$ and ${\overrightarrow{v}}_2=(0,1,-1,3)$.

   1. Find a system of two linear equations in $x_1,x_2,x_3,x_4$ whose solution set is exactly $W$.
   2. Use your equations to decide which of $(3,-1,7,-6)$ and $(1,1,1,1)$ lies in $W$.
2. Q2The Span of a Set of Vectors

   In $P_2$, the space of polynomials of degree at most $2$ with real coefficients, let $W=span\{1+x,x+x^2\}$.

   1. Show that every $p\in W$ satisfies $p(-1)=0$.
   2. Show conversely that every $p\in P_2$ with $p(-1)=0$ lies in $W$, by writing such a $p$ explicitly as a linear combination of $1+x$ and $x+x^2$.
   3. Conclude that $W=\{p\in P_2\mid p(-1)=0\}$, and give one polynomial of $P_2$ that is not in $W$.
3. Q3Testing Membership in a Span

   In ${ℝ}^4$ let ${\overrightarrow{u}}_1=(1,-1,0,2)$, ${\overrightarrow{u}}_2=(0,1,2,-1)$ and ${\overrightarrow{u}}_3=(2,0,3,1)$.

   1. Decide whether $\overrightarrow{b}=(4,-3,1,6)$ lies in $span\{{\overrightarrow{u}}_1,{\overrightarrow{u}}_2,{\overrightarrow{u}}_3\}$. If it does, give the weights.
   2. Decide whether ${\overrightarrow{e}}_1=(1,0,0,0)$ lies in the same span.
4. Q4Testing Membership in a Span

   In $P_3$, the space of polynomials of degree at most $3$, let $p_1=1+x^3$, $p_2=x-x^2$ and $p_3=1+x+x^2+x^3$.

   1. Show that $r=3+x+3x^2+3x^3$ is a linear combination of $p_1,p_2,p_3$, and give the weights.
   2. Show that $q=2+x+3x^2+5x^3$ is not.
5. Q5Linear Independence in n-Space

   Decide whether each set in ${ℝ}^4$ is linearly independent. For a dependent set, give an explicit dependence relation.

   1. $\{(1,2,-1,0),(0,1,3,1),(1,4,5,2)\}$
   2. $\{(1,1,0,2),(0,1,1,1),(2,0,-1,3),(1,0,1,0)\}$
6. Q6Linear Independence in n-Space

   Decide, without row reducing anything, whether each set is linearly independent, and give the reason in one sentence.

   1. $\{(1,0,2,1),(0,3,1,-1),(2,2,0,5),(1,1,1,1),(4,0,-3,2)\}$ in ${ℝ}^4$
   2. $\{(3,1,0,-2,5),(0,0,0,0,0),(1,4,-1,0,2)\}$ in ${ℝ}^5$
   3. $\{(2,-4,6,0,8),(-3,6,-9,0,-12)\}$ in ${ℝ}^5$
   4. $\{(1,0,0,0,5),(0,0,3,0,0)\}$ in ${ℝ}^5$
7. Q7Linear Independence of Polynomials, Matrices and Functions

   Decide whether each set is linearly independent. For a dependent set, give an explicit dependence relation.

   1. In $P_3$ (polynomials of degree at most $3$): $\{1+x+x^2,x+x^2+x^3,1-x^3\}$.
   2. In $M_{2\times 2}$ (real $2\times 2$ matrices): $\left\{\right[\begin{array}{cc}1& 2\\ 0& -1\end{array}],[\begin{array}{cc}0& 1\\ 3& 1\end{array}],[\begin{array}{cc}2& 1\\ -1& 0\end{array}],[\begin{array}{cc}1& 0\\ 1& 1\end{array}\left]\right\}$.
8. Q8Linear Independence of Polynomials, Matrices and Functions

   Here functions $ℝ\to ℝ$ are vectors, and a set $\{f_1,\dots ,f_k\}$ is linearly dependent when there are real numbers $c_1,\dots ,c_k$, not all zero, with $c_1{f}_1(x)+\dots +c_k{f}_k(x)=0$ for every real $x$.

   1. Show that $\{\sin (x+\frac{\pi }{3}),\sin x,\cos x\}$ is linearly dependent by giving an explicit relation. You may use $\sin (A+B)=\sin A\cos B+\cos A\sin B$.
   2. Show that $\{1,x,e^x\}$ is linearly independent, by evaluating a vanishing linear combination at $x=0$, $x=1$ and $x=2$.
9. Q9Dependence Relations and Redundant Vectors

   In ${ℝ}^4$ consider the list $${\overrightarrow{v}}_1=(1,2,0,1),{\overrightarrow{v}}_2=(2,4,0,2),{\overrightarrow{v}}_3=(0,1,1,-1),{\overrightarrow{v}}_4=(1,3,1,0),{\overrightarrow{v}}_5=(0,0,1,1).$$

   1. Find every vector in the list that is a linear combination of the vectors before it, and write each such combination explicitly.
   2. Name three vectors of the list whose span is $span\{{\overrightarrow{v}}_1,\dots ,{\overrightarrow{v}}_5\}$, and say why nothing is lost.
10. Q10Synthesis — drawing on several topics in this unit

    In $M_{2\times 2}$, the space of real $2\times 2$ matrices, let $$A_1=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right],A_2=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],A_3=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right],W=span\{A_1,A_2,A_3\}.$$

    1. Show that $\{A_1,A_2,A_3\}$ is linearly dependent, and give a dependence relation.
    2. Show that $W=\left\{\right[\begin{array}{cc}a& b\\ c& d\end{array}\left]\right|d=a,c=a+b\}$.
    3. Decide whether $B=\left[\begin{array}{cc}2& 3\\ 5& 2\end{array}\right]$ and $I_2$ lie in $W$; for any that does, write it as a combination of $A_1$ and $A_2$.

The 7 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 Span and Linear Independence notes cover  About the Span and Linear Independence 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.

A span, described by equations

The span of a set is every linear combination of its vectors. As a list of combinations it is hard to test against; as the solution set of a few equations it is easy. Q1 asks you to turn the span of two vectors in four coordinates into such a system.

Part (b) is the reward: membership becomes substitution into two equations. Q2 does the same in the quadratic polynomials, in two directions. Part (a) checks that every member of the span has the property; part (b) shows every polynomial with the property is a member, by producing the weights. Together they prove two sets are equal, which is how set equality is always proved.

Testing membership in a span

Q3 and Q4 ask the direct question: is this vector a linear combination of these? That is always a linear system whose unknowns are the weights and whose columns are the given vectors.

Q4 does the same thing with cubic polynomials. Match coefficients of 1, x, x² and x³ to get four equations in three weights. Part (b) asks you to show a combination is impossible, and the proof is the inconsistent row itself — point to it.

Linear independence in n-space

A set is linearly independent when the only combination giving the zero vector is the one with every weight zero. Q5 tests two sets with that definition: the homogeneous system with the vectors as columns has only the trivial solution exactly when every column has a pivot.

Q6 asks you to decide four cases without row reducing, with a one-sentence reason each. The reasons are facts the course proves once and then uses everywhere:

Independence of polynomials, matrices and functions

Q7 repeats Q5 in two other spaces. For polynomials, a combination is the zero polynomial when every coefficient is zero, so matching the coefficients of each power of x gives the homogeneous system. For 2 × 2 matrices, each of the four entries gives one equation.

Q8 is about functions, where there are no coordinates to match, and the question states the definition carefully: a relation must hold for every real x. Part (a) proves dependence with an identity — expand the first function with the formula given and look for a combination of the other two. Part (b) proves independence by evaluation.

Redundant vectors in a list

Q9 gives five vectors and asks which are combinations of the ones before them. Row reduce the matrix whose columns are the five vectors, in order: a column without a pivot is a combination of the earlier pivot columns, and the reduced form shows the weights. Part (b) follows: removing a vector that is a combination of the others does not change the span, so the pivot columns alone span the same space.

The synthesis question

Q10 puts the set together in the 2 × 2 matrices. Part (a) is Q7(b): a homogeneous system from the entries, then an explicit relation. Part (b) is Q1 and Q2 combined — describe the span by conditions on the entries, and show both inclusions. Part (a) helps here, because the relation shows one of the three matrices is redundant. Part (c) is Q3: test each matrix against the conditions, and for any that passes, find its weights on the two matrices that remain.

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Getting the most out of it

Put the vectors in as columns

Membership, independence and redundancy are all questions about the matrix whose columns are the given vectors. Build it the same way every time, in the order the vectors are listed, and the three questions become three readings of one echelon form.

Know which reading answers which question

Consistency of the augmented system answers membership. A pivot in every column answers independence. The non-pivot columns answer redundancy. Before you reduce, write down which of these you are looking for.

Always check a relation

Any explicit combination you produce — weights for membership, or a dependence relation — can be checked in a few seconds by computing it. Do it every time; it catches most arithmetic slips.

Say which shortcut you used

When a quick fact settles independence, name it in your answer. "Too many vectors" and "contains the zero vector" are complete reasons; "obviously dependent" is not.

Common questions