Lesson test: Linear transformations

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Linear transformations

1.[2p]
What do the columns of a matrix contain?

The images of the basis vectors under the transformation

Correct

The coefficients of the first equation of the system

The answer is: The images of the basis vectors under the transformation

The eigenvalues of the transformation

The answer is: The images of the basis vectors under the transformation

The coordinates of a single point being moved

The answer is: The images of the basis vectors under the transformation

2.[3p]
Which of these maps from $R^{2}$ to $R^{2}$ are linear?

Select all that apply

$T (x, y) = (2 x - y, x + 3 y)$

Correct

$T (x, y) = (0, 0)$

Correct

$T (x, y) = (x + 1, y)$

The answer is: $T(x, y) = (2x - y,\ x + 3y)$, $T(x, y) = (0, 0)$

$T (x, y) = (x + y, x y)$

The answer is: $T(x, y) = (2x - y,\ x + 3y)$, $T(x, y) = (0, 0)$

3.[1p]
Every linear map sends the zero vector to the zero vector.

True

Correct

False

The answer is: True

4.[2p]
With $A = [(2, 1), (- 1, 3)]$ and $x = (4, 2)$ , what is the first component of $A x$ ?

CorrectNot quite: 6

5.[2p]
A linear map sends $(1, 0)$ to $(3, 1)$ and $(0, 1)$ to $(- 1, 2)$ . What is the second component of its value at $(4, 5)$ ?

CorrectNot quite: 14

6.[3p]
Match each matrix, written as its list of columns, to the transformation it performs.

$[(1, 0), (0, 1)]$
$[(0, 1), (- 1, 0)]$
$[(1, 0), (0, 0)]$
$[(1, 0), (1, 1)]$

a shear that slides points sideways with height
leaves every vector alone
projection onto the horizontal axis
a quarter turn anticlockwise

Show the answer

$[(1, 0), (0, 1)]$ : leaves every vector alone $[(0, 1), (- 1, 0)]$ : a quarter turn anticlockwise $[(1, 0), (0, 0)]$ : projection onto the horizontal axis $[(1, 0), (1, 1)]$ : a shear that slides points sideways with height

7.[2p]
Why is translation excluded from the linear maps?

It moves the origin, and a linear map must fix it

Correct

It does not send straight lines to straight lines

The answer is: It moves the origin, and a linear map must fix it

It cannot be written with numbers

The answer is: It moves the origin, and a linear map must fix it

It fails only in dimensions above two

The answer is: It moves the origin, and a linear map must fix it

8.[2p]
Rotating $(2, 0)$ by $30 °$ gives a vector whose first component is what, to three decimal places?

CorrectNot quite: 1.732

9.[2p]
An $m$ by $n$ matrix represents a map

from $R^{n}$ to $R^{m}$

Correct

from $R^{m}$ to $R^{n}$

The answer is: from $\mathbb{R}^n$ to $\mathbb{R}^m$

from $R^{n}$ to $R^{n}$ only

The answer is: from $\mathbb{R}^n$ to $\mathbb{R}^m$

only when $m = n$

The answer is: from $\mathbb{R}^n$ to $\mathbb{R}^m$

1.[2p] What do the columns of a matrix contain?

2.[3p] Which of these maps from R2 to R2 are linear?

3.[1p] Every linear map sends the zero vector to the zero vector.

4.[2p] With A=[(2,1),(-1,3)] and x=(4,2), what is the first component of Ax?

5.[2p] A linear map sends (1,0) to (3,1) and (0,1) to (-1,2). What is the second component of its value at (4,5)?

6.[3p] Match each matrix, written as its list of columns, to the transformation it performs.

7.[2p] Why is translation excluded from the linear maps?

8.[2p] Rotating (2,0) by 30° gives a vector whose first component is what, to three decimal places?

9.[2p] An m by n matrix represents a map

1.[2p]
What do the columns of a matrix contain?

2.[3p]
Which of these maps from $R^{2}$ to $R^{2}$ are linear?

3.[1p]
Every linear map sends the zero vector to the zero vector.

4.[2p]
With $A = [(2, 1), (- 1, 3)]$ and $x = (4, 2)$ , what is the first component of $A x$ ?

5.[2p]
A linear map sends $(1, 0)$ to $(3, 1)$ and $(0, 1)$ to $(- 1, 2)$ . What is the second component of its value at $(4, 5)$ ?

6.[3p]
Match each matrix, written as its list of columns, to the transformation it performs.

7.[2p]
Why is translation excluded from the linear maps?

8.[2p]
Rotating $(2, 0)$ by $30 °$ gives a vector whose first component is what, to three decimal places?

9.[2p]
An $m$ by $n$ matrix represents a map