University Linear Algebra — Linear Transformations Worksheet

All 14 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. Q1Verifying That a Map Is Linear

   A map $T:V\to W$ between vector spaces is linear when $T(\overrightarrow{u}+\overrightarrow{v})=T(\overrightarrow{u})+T(\overrightarrow{v})$ and $T(c\overrightarrow{u})=cT(\overrightarrow{u})$ for all $\overrightarrow{u},\overrightarrow{v}\in V$ and every real scalar $c$. Decide whether each map is linear. For a linear map, verify both properties; for a map that is not, give specific vectors (and a scalar, if needed) at which a property fails.

   1. $T:{ℝ}^4\to {ℝ}^2$, $T(x_1,x_2,x_3,x_4)=(x_1-2x_3,x_2+x_4)$
   2. $S:{ℝ}^4\to {ℝ}^2$, $S(x_1,x_2,x_3,x_4)=(x_1+1,x_3)$
   3. $U:{ℝ}^4\to {ℝ}^2$, $U(x_1,x_2,x_3,x_4)=(x_1{x}_2,x_4)$
2. Q2Verifying That a Map Is Linear

   Let $P_2$ be the vector space of polynomials of degree at most $2$, and define $T:P_2\to P_2$ by $$T(p(x))=p(x+1)-p(x).$$

   1. Show that $T$ is linear.
   2. Compute $T(3-x+2x^2)$.
3. Q3The Standard Matrix of a Linear Transformation

   Let $T:{ℝ}^4\to {ℝ}^3$ be the linear transformation $$T(x_1,x_2,x_3,x_4)=(x_1+2x_2-x_4,3x_2+x_3,x_1-x_3+4x_4).$$

   1. Find the standard matrix $A$ of $T$, so that $T(\overrightarrow{x})=A\overrightarrow{x}$.
   2. Use $A$ to compute $T(1,-1,2,0)$.
4. Q4The Standard Matrix of a Linear Transformation

   A linear transformation $T:{ℝ}^2\to {ℝ}^4$ satisfies $$T(1,1)=(3,1,0,2)\text{and}T(1,-1)=(1,-1,4,0).$$ Find the standard matrix of $T$.

5. Q5Rotations, Reflections, Projections and Shears

   Find the standard matrix of each linear operator on ${ℝ}^2$, then answer the question that follows it.

   1. $R$, the counterclockwise rotation about the origin through $\frac{2\pi }{3}$. Find $R(2,2\sqrt{3})$.
   2. $H$, the shear parallel to the $x$-axis that sends $(0,1)$ to $(-2,1)$ and fixes every point of the $x$-axis. Find the image under $H$ of the unit square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$, and its area.
6. Q6Rotations, Reflections, Projections and Shears

   Let $L$ be the line $y=2x$ in ${ℝ}^2$.

   1. Find the standard matrix $P$ of the orthogonal projection onto $L$, and verify that $P^2=P$.
   2. Find the standard matrix $F$ of the reflection in $L$, and verify that $F^2=I_2$.
   3. Find the projection and the reflection of $(5,0)$.
7. Q7The Kernel and Range of a Linear Transformation

   Let $T:{ℝ}^4\to {ℝ}^3$ be $T(\overrightarrow{x})=A\overrightarrow{x}$, where $$A=\left[\begin{array}{cccc}1& 2& 0& -1\\ 2& 4& 1& 1\\ -1& -2& 1& 4\end{array}\right].$$

   1. Find a basis for the kernel $\ker T=\{\overrightarrow{x}\in {ℝ}^4\mid T(\overrightarrow{x})=\overrightarrow{0}\}$.
   2. Find a basis for the range $imT=\{T(\overrightarrow{x})\mid \overrightarrow{x}\in {ℝ}^4\}$, and describe it by a single linear equation in $(y_1,y_2,y_3)$.
8. Q8The Kernel and Range of a Linear Transformation

   Let $P_3$ be the vector space of polynomials of degree at most $3$, and let $T:P_3\to {ℝ}^2$ be the linear map $T(p)=(p(0),p(2))$. Find a basis for $\ker T$ and a basis for $imT$.

9. Q9The Dimension Theorem for Linear Transformations

   Let $M_{2\times 2}$ be the vector space of real $2\times 2$ matrices, and let $T:M_{2\times 2}\to M_{2\times 2}$ be the linear map $T(A)=A-A^T$.

   1. Describe $\ker T$ and find its dimension.
   2. Describe $imT$ and find its dimension.
   3. Check that your answers agree with the dimension theorem.
10. Q10The Dimension Theorem for Linear Transformations

    Let $T:{ℝ}^7\to {ℝ}^5$ be linear. Use the dimension theorem, $\dim \ker T+\dim imT=7$, to answer each part.

    1. If $T$ is onto ${ℝ}^5$, what is $\dim \ker T$?
    2. If $\dim \ker T=4$, what is $\dim imT$, and is $T$ onto?
    3. What is the smallest possible value of $\dim \ker T$? Can $T$ be one-to-one?
11. Q11One-to-One and Onto Transformations

    For each linear map, decide whether it is one-to-one and whether it is onto, and justify each verdict with the kernel or the dimension of the range.

    1. $T:{ℝ}^3\to {ℝ}^4$, $T(x_1,x_2,x_3)=(x_1,x_1+x_2,x_2+x_3,x_3)$.
    2. $E:P_2\to {ℝ}^3$, $E(p)=(p(0),p(1),p(2))$, where $P_2$ is the space of polynomials of degree at most $2$.
12. Q12Composition of Linear Transformations

    Let $T:{ℝ}^4\to {ℝ}^2$ and $S:{ℝ}^2\to {ℝ}^3$ be the linear maps $$T(x_1,x_2,x_3,x_4)=(x_1+x_2,x_3-x_4),S(y_1,y_2)=(y_1,y_1+y_2,2y_2).$$

    1. Find the standard matrix of $S\circ T$, and state its domain and codomain.
    2. Compute $(S\circ T)(1,2,3,4)$ two ways: with the matrix, and by applying $T$ and then $S$.
    3. Is $T\circ S$ defined? Explain.
13. Q13Invertible Transformations and Isomorphism

    Let $M_{2\times 2}$ be the space of real $2\times 2$ matrices and $P_3$ the space of polynomials of degree at most $3$. Define $T:M_{2\times 2}\to P_3$ by $$T\left(\right[\begin{array}{cc}a& b\\ c& d\end{array}\left]\right)=a+(a+b)x+(b+c)x^2+(c+d)x^3.$$

    1. Show that $T$ is invertible by finding a formula for $T^{-1}$.
    2. Find the matrix $A$ with $T(A)=1+3x+2x^2-x^3$.
14. Q14Synthesis — drawing on several topics in this unit

    Let $P_3$ be the space of polynomials of degree at most $3$, and define $T:P_3\to {ℝ}^3$ by $$T(p)=(p(1),p^{\prime }(1),p^{″}(1)).$$

    1. Show that $T$ is linear.
    2. Find a basis for $\ker T$.
    3. Use the dimension theorem to find $\dim imT$. Is $T$ onto? Is it one-to-one?

The 10 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 Linear Transformations notes cover  About the Linear Transformations 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 map linear?

Q1 gives three maps from four-space to the plane and asks which are linear. The definition is in the question: a linear map respects addition and scalar multiplication. Before verifying anything, decide.

Q2 is a map on quadratic polynomials that shifts the input and subtracts. Part (a) is a general verification: substitute p + q and cp into the rule and use the fact that evaluating a sum of polynomials at a point is the sum of the values. Part (b) is a computation — expand the shifted polynomial carefully before subtracting.

The standard matrix

Every linear map from Rⁿ to Rᵐ is multiplication by an m × n matrix, and its columns are the images of the standard basis vectors.

In Q3 the rule is given by a formula, so each row of A holds the coefficients of one output coordinate — check the size first: three outputs, four inputs. Q4 gives the images of two vectors that are not the standard ones. Write each standard basis vector as a combination of the two given vectors, then use linearity to find its image.

Rotations, reflections, projections and shears

Q5 and Q6 are the geometric transformations of the plane, each built from where it sends the two standard basis vectors. For the rotation in Q5(a), the images of e₁ and e₂ come from the cosine and sine of the angle — work out both exactly before assembling. For the shear in Q5(b), the question tells you where e₂ goes, and e₁ is fixed because the x-axis is.

Q6 builds the projection onto a line that passes through the origin, and the reflection in it. The projection comes from the CEGEP formula for projecting a vector onto a direction vector of the line, applied to e₁ and e₂. The reflection comes from the projection: a point and its reflection have the projection as their midpoint.

Kernel and range

The kernel of T is everything it sends to zero; the range is everything it produces. When T is multiplication by A, these are exactly A's null space and column space, and Q7 is the previous set's method under new names: reduce A, read a null-space basis from the free variables and a column-space basis from the pivot columns of A.

Q7(b) also asks for the range as a single equation. Augment A with a general target (y₁, y₂, y₃) and reduce; the row whose coefficient part vanishes gives the condition on the y's, as in the span set.

Q8 asks the same questions of a map from cubic polynomials to the plane, given by evaluation at two points. For the kernel, write the conditions p(0) = 0 and p(2) = 0 as equations in the four coefficients and solve. For the range, ask which pairs of values a cubic can take at two points — and let the dimension theorem confirm your answer.

The dimension theorem

Q9 uses it on the map sending a 2 × 2 matrix A to A minus its transpose. Part (a) asks when the output is zero — which matrices equal their own transpose. Part (b) asks what outputs are possible; compute the image of a general matrix and see which entries are forced. Part (c) is the check that the two dimensions add up correctly.

Q10 is the theorem used on its own, for a map from seven-space to five-space about which you know nothing else. Each part is one line: substitute what you are told into the theorem and compare with the dimension of the codomain.

One-to-one and onto

For a linear map, one-to-one means the kernel is only the zero vector, and onto means the range is the whole codomain. Q11 asks for both verdicts on two maps, each justified by a kernel or a dimension.

Composition is matrix multiplication

Q12 composes a map from four-space to the plane with a map from the plane to three-space. Doing T first and then S corresponds to the product of their standard matrices with S's on the left.

Part (b) is a check that the product is right: the two computations must agree. Part (c) asks whether the composition in the other order is defined — compare the codomain of S with the domain of T.

Invertible maps and isomorphism

Q13 defines a map from the 2 × 2 matrices to the cubic polynomials. Both spaces have dimension four, which is what makes an invertible map between them possible. To find the inverse, set the output equal to a general cubic, match coefficients, and solve for a, b, c and d in terms of the cubic's coefficients — the system here unwinds one variable at a time. Part (b) applies the inverse you found to a specific polynomial.

The synthesis question

Q14 takes one map from cubic polynomials to three-space, built from a value and two derivatives at the same point. Part (a) is Q2's verification, using the fact that differentiation respects sums and constant multiples. Part (b) is Q8's method: write the three conditions as equations in the four coefficients, or think about which cubics have value, slope and second derivative all zero at that point. Part (c) is Q10 and Q11: the theorem gives the dimension of the range, and comparing it with the codomain and the kernel with zero gives the two verdicts.

Preview all 8 pages

Click any page to open the full PDF.

Getting the most out of it

Write domain and codomain every time

Before any computation, write T : V → W with both spaces and their dimensions. The size of the matrix, the dimension theorem, and the one-to-one and onto questions all start from those two numbers.

Compute the image of zero first

It is the fastest linearity test there is. A map that does not send zero to zero is not linear, and you are done.

Build matrices column by column

The standard matrix is the images of e₁, e₂, … as columns. Say that sentence as you build each one — for a rotation, a shear or a projection it keeps you from transposing the matrix by accident.

Let the dimension theorem do the second half

Once you have the kernel, the dimension of the range is free. Use the theorem to find it, then find an actual basis of the range as a check.

Common questions