Matrix Worksheet - Math 205

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Math 205 Quiz 3
Name:
1. Consider the linear transformation T (x
, x
, x
) = (x
2x
, x
+ x
+ x
).
1
2
3
1
2
1
2
3
3
(a) What is the domain of T ?
2
(b) What is the codomain of T ?
1
(c) What is the image of x =
2
?
3
3
6
(d) Determine the matrix A such that T (x) = Ax.
T ( e
) = (1, 1), T ( e
) = ( 2, 1), T ( e
) = (0, 1)
1
2
3
1
2 0
A =
1
1
1
(e) Is T one-to-one? Briefly explain.
2
No, the matrix A has a free variable, so there may be infinite many x sent to a b in
.
(f) Is T onto? Briefly explain.
2
Yes, the matrix A has a pivot in every row. So the columns of A span
.
2. Provide a brief written answer to the following.
(a) The matrix equation Ax = b is inconsistent if and only if rref([A b])
There is a row [000000 1] or a pivot in the last column..
(b) What is the definition of the span of a set of vectors?
The span is the set of all linear combinations of the vectors.
(c) If a set of vectors is linearly dependent, then what does that mean? NOT how can you tell,
what does it mean?
That means that at least one of the vectors is a linear combination of the others.
(d) How can you tell if a set of vectors is linearly dependent?
You can solve the system Ax = 0 and look for a free variable. This means there exists a non-trivial
solution and you can write a linear dependence relation.
(e) Give one statement that is equivalent to: “Let A be an m
n matrix. The matrix equation
Ax = b has a solution for every b in
.”
i. Every row of A has a pivot.
ii. The columns of A span
.
iii. Every b in
can be written as a linear combination of columns of A.
iv. The transformation T (x) = Ax is onto.
1

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