Page 24 - 35Linear Algebra

24 What is Linear Algebra?

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2 6 x x 2 6 x 2 6 x
2. + = + .
4 8 y y 0 4 8 y 4 8 y 0
These equalities can be verified using the rules we introduced so far.

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Example 8 Verify that is a linear operator.
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The matrix-function is homogeneous if the expressions on the left hand side and right
hand side of the first equation are indeed equal.

2 6 a 2 6 λa 2 6
λ = = λa + λb
4 8 b 4 8 λb 4 8

2λa 6bc 2λa + 6λb
= + =
4λa 8bc 4λa + 8λb

while

2 6 a 2 6 2a 6b
λ = c a + b = λ +
4 8 b 4 8 4a 8b

2a + 6b 2λa + 6λb
= λ = .
4a + 8b 4λa + 8λb
The underlined expressions are identical, so the matrix is homogeneous.

The matrix-function is additive if the left and right side of the second equation are
indeed equal.

2 6 a c 2 6 a + c 2 6
+ = = (a + c) + (b + d)
4 8 b d 4 8 b + d 4 8

2(a + c) 6(b + d) 2a + 2c + 6b + 6d
= + =
4(a + c) 8(b + d) 4a + 4c + 8b + 8d
which we need to compare to

2 6 a 2 6 c 2 6 2 6
+ = a + b + c + d
4 8 b 4 8 d 4 8 4 8

2a 6b 2c 6d 2a + 2c + 6b + 6d
= + + + = .
4a 8b 4c 8d 4a + 4c + 8b + 8d
Thus multiplication by a matrix is additive and homogeneous, and so it is, by definition,
linear.

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