Linear Algebra
Name:
Quiz 5
Id No.:
Class:
Problem 1: Let V and W be vector spaces, and let T and U be nonzero linear
transformations from V into W . If R(T ) R(U ) = cfw_0, prove that cfw_T, U is a
linearly independent subset of L(V, W ). (8 points)
Pro
Linear Algebra
Solution to Midterm 1
Problem 1: Let V = cfw_(a1 , a2 ) : a1 , a2 2 R. Define addition of elements of V coordinatewise, and for (a1 , a2 ) in V and c 2 R, define
n(0,0)
if c=0
c(a1 , a2 ) =
(ca1 , ac2 ) if c6=0.
Is V a vector space over R w
Math 24
Spring 2012
Sample Homework Solutions
Week 4
Section 2.3
(3) Let g(x) = 3 + x. Let T : P2 (R) P2 (R) and U : P2 (R) R3 be the
linear transformations respectively defined by
T (f (x) = f 0 (x)g(x) + 2f (x)
U (a + bx + cx2 ) = (a + b, c, a b).
Let a
Math 24
Spring 2012
Sample Homework Solutions
Week 3
In-class problems from Monday, April 9:
(2) Give examples of pairs of subspaces W1 and W2 of R3 , neither of which
is contained in the other, such that:
(a) W1 + W2 6= R3 . In your example, what is W1 +
Math 24
Spring 2012
Sample Homework Solutions
Week 4
Section 2.3
(3) Let g(x) = 3 + x. Let T : P2 (R) P2 (R) and U : P2 (R) R3 be the
linear transformations respectively defined by
T (f (x) = f 0 (x)g(x) + 2f (x)
U (a + bx + cx2 ) = (a + b, c, a b).
Let a
Honors Linear Algebra-Homework 1
Yuanqing Cai
ycai7@jhu.edu
1
1.1 Introduction
2.Find the equations of the lines through the following pairs of points in
space.
(a)(3, 1, 4) and (5, 7, 1)
(b)(2, 4, 0) and (3, 6, 0)
(c)(3, 7, 2) and (3, 7, 8)
(d)(2, 1, 5)
Math 115A HW3 Solutions
August 29, 2012
1
University of California, Los Angeles
Problem 2.2.10
Let V be a vector space with the ordered basis = cfw_v1 , ., vn . Define v0 = 0. By Theorem 2.6,
there exists a linear transformation T : V V such that T (vj )
Honors Linear Algebra-Homework 3
Yuanqing Cai
ycai7@jhu.edu
1
1.4 Linear Combinations and Systems of Linear Equations
3.For each of the following lists of vectors in R3 , determine whether the first
vector can be expressed as a linear combination of the o
Practice Problems
MTH 5102
3/13/2007
1. Let and be the standard bases for IRn and IRm respectively. For each
linear transformation T : IRn IRm given below, compute [T ] .
(i) T (a1 , a2 , a3 ) = (2a"1 + 3a2 a3 ,#a1 + a3 ). (ii)T (a1 , a2 , a3 ) = 2a1 + a2
Math 24
Spring 2012
Thursday, April 19
(1.) TRUE or FALSE?
(a.) Suppose that = cfw_x1 , x2 , . . . , xn and 0 = cfw_x01 , x02 , . . . , x0n are ordered bases for
a vector space and Q is the change of coordinate matrix that changes 0 -coordinates into
-c
Math 115A
Homework 4 Comments
I graded 8 of the problems:
Section 2.2: 4, 8, 10, 13
Section 2.3: 2b, 4b, 11, 17
Each problem is worth 2 points. A grade of 0 indicates no solution or a substantially wrong solution. A
grade of 2 indicates a correct or nearl
Linear Algebra Take Home Test 4
7/16/2009
Solutions
1. Let V and W be finite dimensional vector spaces and suppose T : V W is linear. Prove that
T is injective if and only if dim(V ) = rank(T).
Proof: () Suppose T is injective, then ker T = cfw_0, so Null
Partial Solutions for Linear Algebra by Friedberg et al.
Chapter 1
John K. Nguyen
December 7, 2011
1.1.8. In any vector space V , show that (a + b)(x + y) = ax + ay + bx + by for any x, y 2 V and any a, b 2 F .
Proof. Let x, y 2 V and a, b 2 F . Note that