By Aaron Marcus

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**Extra info for An Introduction to Homological Algebra**

**Example text**

3. Let v1 = (1, −1, 1) and v2 = (1, −2, 2). Find a vector v3 , such that f (v3 ) = v2 + v3 , and prove that (v1 , v2 , v3 ) form a basis of R3 . 4. Find the matrix of f with respect to the basis (v1 , v2 , v3 ). 1. We ﬁrst compute det(A − λI) = = 4 5−λ −4 4 1 1−λ 1 1−λ 0 −1 2 −1 − λ −1 2 − (λ + 1) 5−λ −4 1 1−λ = 4{2 + 1 − λ} − (λ + 1){(λ − 1)(λ − 5) + 4} = −4(λ − 3) − (λ + 1){λ2 − 6λ + 9} = −4(λ − 3) − (λ + 1)(λ − 3)2 = −(λ − 3){(λ + 1)(λ − 3) + 4} = −(λ − 3){λ2 − 2λ + 1} = −(λ − 1)2 (λ − 3). com 32 Linear Algebra Examples c-3 1.

The complete solution is x1 (t) x2 (t) = 3e−t −e−t e7t 2e7t c1 c2 , where c1 and c2 are arbitrary constants. 2. We get for t = 0 that x1 (0) x2 (0) = 3 1 −2 2 c1 c2 1 10 = , where 3 1 −2 2 1 10 ∼ 1 3 −2 2 ∼ 1 3 0 1 11 10 11 4 ∼ ∼ 1 0 1 0 0 1 3 8 11 32 −1 4 , hence c1 = −1 and c2 = 4, corresponding to the solution x1 (t) x2 (t) = 3e−t −2e−t e7t 2e7t −1 4 = −3e−t + 4e7t 2e−t + 8e7t . 2 2. Systems of differential equations 1. Solve the systems of diﬀerential equations dx1 (t) dt dx2 (t) dt dx3 (t) dt = 4x1 (t) + x3 (t), = −2x1 (t) + x2 (t) = −2x1 (t) + x3 (t).

If λ = 1, then we get by reduction ⎛ ⎞ ⎛ ⎞ 0 0 0 1 0 1 A − λI = A − I = ⎝ 1 0 1 ⎠ ∼ ⎝ 0 0 0 ⎠ 1 0 1 0 0 0 of rank 1, so the eigenspace has dimension 3 − 1 = 2 with the linearly independent eigenvectors v1 = (1, 0, −1) and v2 = (0, 1, 0). If λ = 2, then we get by reduction ⎛ ⎞ ⎛ ⎞ −1 0 0 1 0 0 A − λI = A − 2I = ⎝ 1 −1 1 ⎠ ∼ ⎝ 0 1 −1 ⎠ 1 0 0 0 0 0 of rank 2, hence the eigenspace has dimension 3 − 2 = 1, and an eigenvector is v 3 = (0, 1, 1). 2. It is immediately seen that ⎛ V = (v1 v2 ⎞ 1 0 0 v3 ) = ⎝ 0 1 1 ⎠ −1 0 1 ⎛ ⎞ 1 0 0 and Λ = ⎝ 0 1 0 ⎠ .

### An Introduction to Homological Algebra by Aaron Marcus

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