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MATH 2331 Lecture Notes - Linear Combination

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12 Feb 2014

School

Northeastern University

Department

Course

Professor

Rita Jimenez Rolland

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Document Summary

Linear combinations of vectors x1(cid:126)v1 + x2(cid:126)v2 = (cid:126)v3 (cid:126)v3 is a linear combination of (cid:126)v1 and (cid:126)v2. 1 x1(cid:126)v1 + x2 (cid:126)v2 = (cid:126)v3 (cid:126)v3 as a linear combination: x1 = 2 x2 = 1. A(cid:126)x: an m(cid:126)xm 1 a(cid:126)xn 1 (cid:126)w1 (cid:126)x (cid:126)w2 (cid:126)x. (cid:126)wn (cid:18) 3 (cid:19) (cid:19)(cid:18) 2 3 (cid:18)0 2 + 1 3 (cid:19) (cid:19) 1 1 + 1 2 + 1 3. 5 1 + 1 2 + 1 3. 1 1 + 5 2 + 3 3 (cid:126)wn (cid:126)x.

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Related Questions

K(x,y,z) = (x, x + y,x + y + z)

L(x,y,z) = (2x - y, x + 2y)

S(x,y,z) = (z,y,x)

T(x,y) = (2x + y,x + y,x - y,x - 2y)

G: R^3 -> M(2,2) defined by G(x,y,z) = (y z , (-x + y) -x)

H: M(2,2) -> P2 defined by H(a b, c d) = (a + b) + (a - 2d)x+ dx^2

Find the indicated linear transformation if it is defined. ifnot defined explain why it is not:

1) K + S

2) GS

3)2S^3 - S^2 + 3S - 4I

Find the matrix that represent each of the linear transformationK,L,S,and T for the following

1)TL

2)K + S

Find the matrix that represent each of linear transformation intwo ways, first using the result normal way and using therepresention of the matrix.

1)

T:R^3 --> R^4 given by T(X) = Ax, where

A = [1 1 0

2 0 3

-1 1 1

1 1 2]

and v1 = (-3,3,2) v2 = (0,0,0)

2) T: R^4 --> R given by T(x1,x2,x3,x4) = x1 + 2x2 + 3x3 +4x4 and v1 = (1,2,1,-2) v2 = (2,3,0,-2)

3)T: M(2,3) --> P2

P2 = (a11 + a13)x^2 + (a21 - a22)x + a23

v1 = [-1 3 1

2 2 0]

v2 = [2 1 -2

1 1 0]

& determine wheter w1 or w2 is the iamge of thecorresponding linear transformation

w1 = x^2 + 2x + 1,w2 = x - 2.