1.8/1.9 Quiz

Question 1

what do we mean when we say find the image under?

Answer 1

we should be given a matrix, A, and if they are in the same space, We find the transformation, T, when we multiply the matrix by the vectors we were asked about.

If we’re asked abt the vectors u and v, find T(u) and T(v)

What we’re saying here is the Transformation of matrix A by vecotrs u and v

Question 2

if there is a free variable, is the solution or vector we get from the system of equations unique?

Answer 2

no

Question 3

Let A be a 7 x 4 matrix. What must a and b be in order to define T :R^a -> R^b by ​T(x​) = Ax​?

Answer 3

a = 4 columns and b = 7 rows

a should be columns and b should be rows

Question 4

what does it mean for something to be mapped onto the zero vector?

Answer 4

Ax = 0

Question 5

how to know of something (b) is in the range of the linear transformation of x |-> Ax? A is a matrix

Answer 5

the augmented matrix [A b] or Ax=b must be consistent

Question 6

what is Ax = b intuitively from the perspective of linear transformations?

Answer 6

multiplication by A transforms x into b

Question 7

domain vs codomain vs range

Answer 7

Domain: Set of possible input values.
Codomain: Set of all possible output values.
Range (Image): Set of actual output values produced by the function.

Question 8

if vector a maps vector b what vector should we use for the problem.

Answer 8

vector b.

Question 9

what to do if you’re given a bunch of vectors?

Answer 9

make the standard matrix

Question 10

what are e_1 and e_2

Answer 10

e_1 [0 1] and e_2 [1 0] unless told other wise