Download Math 205A Exam I - Linear Transformations and Matrix Equations and more Exams Linear Algebra in PDF only on Docsity!
Here are facts you may find useful:
For problem 1:
The RREF of
is
For problems 2 and 3:
is row equivalent to
- Let B =
, let c 1 , c 2 , c 3 and c 4 be the column vectors in B, and let T : R
j →
R
q be the linear transformation which has B as its standard matrix.
1A. What are the values of j and q? j = q =
1B. Do the columns c 1 , c 2 , c 3 and c 4 of B form a linearly independent set? Explain your answer in
terms of the definition of linear independence.
1C. Does Bx = b have a solution for every b in R
q ? Explain your answer.
1D. Find all solutions vh of the homogeneous system Bx = 0.
1E. Calculate T
and call the result “a”. a =
1F. Find all solutions of Bx = a (see part 1E).
1G. Is T onto R
q ? Explain.
- Again let D =
. Let the column vectors of D be labeled d 1 , d 2 ,... , d 5
3A. Express d 2 as a linear combination of the other columns of D or explain why this is impossible.
3B. Express d 3 as a linear combination of the other columns of D or explain why this is impossible.
3C. Write the vector 10d 1 + 20d 2 + 30d 3 + 40d 4 + 50d 5 as a linear combination of d 1 , d 2 , d 4 and d 5
(that is, without using d 3 ) or explain why this is impossible.
4A. Suppose that T : R
a → R
b is a transformation. Define what it means for T to be a linear
transformation. Your answer will include phrases like “for all v in... ”, etc.
4B. Suppose T
x
y
z
x − 2 y + 4z
1
0
. Following the approved way we did so in class, show
why T is not a linear transformation. Does either part of the L.T. definition hold?
