Q49E Verify that the kernel of a line... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q49E (page 121)

Verify that the kernel of a linear transformation is closed under addition and scalar multiplication. See Theorem 3.1.6.

Short Answer

Expert verified

The kernel of a linear transformation is closed under addition and scalar multiplication.

Step by step solution

Step 1: Objective

The objective is to verify that the kernel of a linear transformation is closed under addition and scalar multiplication.

Proving kernel of T is closed under addition

$Consider the linear transformation T (\overset{鈫�}{x}) = A \overset{鈫�}{x}, from R^{m} to R^{n}, where \overset{鈫�}{x} is in R^{m} and A \overset{鈫�}{x} is in R^{n} . Let {\overset{鈫�}{x}}_{1} " and {\overset{鈫�}{x}}_{2} " bethekernelofT . ThenT ({\overset{鈫�}{x}}_{1}) = 0 and T ({\overset{鈫�}{x}}_{2}) = 0 . T ({\overset{鈫�}{x}}_{1} + {\overset{鈫�}{x}}_{2}) = T ({\overset{鈫�}{x}}_{1}) + T ({\overset{鈫�}{x}}_{2}) = \overset{鈫�}{0} + \overset{鈫�}{0} = \overset{鈫�}{0} So, {\overset{鈫�}{x}}_{1} + {\overset{鈫�}{x}}_{2} is in the kernel of T . Hence, the kernel of T is closed under addition .$

Proving kernel of T is closed under multiplication

$Let {\overset{鈫�}{x}}_{1} be in the kernel of T and k 鈭� R . ThenT ({\overset{鈫�}{x}}_{1}) = 0 . T (k {\overset{鈫�}{x}}_{1}) = kT ({\overset{鈫�}{x}}_{1}) = k \overset{鈫�}{0} = \overset{鈫�}{0} So, k \overset{鈫�}{x} is in the kernel of T . Hence, the kernel of T is closed under scalar multiplication .$

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91影视

Verify that the kernel of a linear transformation is closed under addition and scalar multiplication. See Theorem 3.1.6.

Short Answer

Step by step solution

Step 1: Objective

Proving kernel of T is closed under addition

Proving kernel of T is closed under multiplication

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