Download Math 205B Exam 01 - Linear Algebra Problems and more Exams Linear Algebra in PDF only on Docsity!
- Let A =
, let b =
b 1 b 2 b 3
(^) and let c =
1A. Find all solutions of Ax = c and write your answer in parametric vector form x = p + vh where p is a particular solution of Ax = c and and vh represents all solutions of the homogeneous equation Ax = 0.
1B. Find any/all conditions on b 1 , b 2 and b 3 which are necessary and sufficient for b to be in the span of the set of column vectors of A.
1C. Give the correct definition: What does it mean to say a set of vectors S = {v 1 , v 2 ,... , vp} is linearly independent?
“We say S is linearly independent if and only if... ” [you complete the definition:]
1D. Do the column vectors of A form a linearly independent set? Explain in terms of the definition of linear independence.
1E. Do the column vectors of A span R^3? Explain.
- Suppose T : Rk^ → Rq^ is a transformation. Define what it means for T to be a linear transformation.
- Suppose T
s 1 s 2 s 3
[
s 12 − s 22 s 1 + s 2 + s 3
]
is a transformation from R^3 → R^2.
Show T is not a linear transformation and in fact fails both parts of the definition in problem (2). In your counterexamples’ vectors, use all-different, positive, single digit numbers for the si’s.
- Suppose that T : R^3 → R^4 is a linear transformation and T
and^ T
5A. Find T
5B. Find T
5C. Let D be the standard matrix of T. How many rows and columns, respectively, does D have?
Number of ROWS: Number of COLUMNS:
5D. Find the last column of D, or explain why you can’t.
5E. Find the first column of D, or explain why you can’t.
- For each separate problem below, find a matrix A in RREF (reduced row echelon form) with exactly 3 rows that satisfies the conditions given in the problem. Create your RREF using as few zeros as possible. If there is no such RREF, explain why.
6A. Ax = b has exactly one solution for every b in R^3.
6B. A is the standard matrix of a linear transformation T : R^2 → R^3 and T is onto R^3.
6C. A has 5 columns and some vectors in R^3 are not in the span of those five column vectors.
6D. A is the standard matrix of a linear transformation T : R^4 → R^3 and T is one-to-one.
