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Applied Linear Algebra - Solved Quiz 29 | MATH 310, Quizzes of Linear Algebra

University of Illinois - ChicagoLinear Algebra

Material Type: Quiz; Class: Applied Linear Algebra; Subject: Mathematics; University: University of Illinois - Chicago; Term: Unknown 2012;

Typology: Quizzes

2011/2012

Uploaded on 05/18/2012

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MATH 310
Self-quiz 29
1. Consider the linear map T:R3โ†’R3with:
T((x, y, z)T)=(xโˆ’y, 2x+y+ 3z, x +y+ 2z)T
Determine the dimension of its kernel and the dimension of its image.
2. Determine whether the vector (1,โˆ’1,2)Tis a linear combination of the
vectors (1,3,2)Tand (2,โˆ’1,1)Tor not.
pf3
pf4

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MATH 310

Self-quiz 29

  1. Consider the linear map T : R^3 โ†’ R^3 with:

T ((x, y, z)T^ ) = (x โˆ’ y, 2 x + y + 3z, x + y + 2z)T

Determine the dimension of its kernel and the dimension of its image.

  1. Determine whether the vector (1, โˆ’ 1 , 2)T^ is a linear combination of the vectors (1, 3 , 2)T^ and (2, โˆ’ 1 , 1)T^ or not.

MATH 310

Self-quiz 29

  1. Consider the linear map T : R^3 โ†’ R^3 with:

T ((x, y, z)T^ ) = (x โˆ’ y, 2 x + y + 3z, x + y + 2z)T

Determine the dimension of its kernel and the dimension of its image.

Solution: We know that the two dimensions add up to 3. So it is enough to compute one of them. The dimension of the image is the same as the rank of the matrix representing this linear map with respect to the canonical bases. This matrix is as follows:

A =

We will compute the rank of this matrix by performing row and column operations that will bring it in a particularly simple form:

โˆ’ 2 R 1 + R 2

โˆ’R 1 + R 3

C 1 + C 2

1 3 R^2

โˆ’ 2 R 1 + R 2

โˆ’R 1 + R 3

โˆ’ 3 C 1 + C 2

โˆ’ 2 C 1 + C 3

R 2 โ†” R 3

โˆ’ 14 R 2

7 R 2 + R 3

โˆ’ 13 R 3

โˆ’ 13 R 3

This means that this matrix has rank 3 and therefore the given vector is not a linear combination of the other two.