# Linear Algebra

** Linear Algebra (Spring 2005, Prof. Aitken).
**Problems 1–2: Non-singular matrices. Let F be a field and let V = F

^{n}. Let : V

^{n}→ F be the unique

normalized alternating n- linear functional on V . (Normalized means ).

1. Let f : V

^{n}→ F be an alternating n-linear functional. Show that if is a linear combination of

then . Conclude that if the columns of a matrix are linearly dependent , then

its determinant is 0. Show that if the rows of a matrix are linearly dependent , then its determinant is 0.

2. Show that if a square matrix has linear independent columns, then A is invertible, and

detA ≠ 0. Hint: see LA10 problems 1 and 2, and LA16 problem 9. Now prove the following:

**Theorem.**Let be a square matrix where F is a field. Then the following are equivalent:

1. The matrix A is invertible. (So . In other words, A is non-singular).

2. detA ≠ 0.

3. The column vectors of A form a basis for F

^{n}.

4. The column vectors of A are linearly independent in F

^{n}.

5. The column vectors of A span F

^{n}.

6. The row vectors of A form a basis of F

^{n}.

7. The row vectors of A are linearly independent in F

^{n}.

8. The row vectors of A span F

^{n}.

Problems 3–6: Cramer’s Rule. Now we know that detA ≠ 0 is equivalent to A being invertible, if the scalars

are a field F. But what if the scalars are a commutative ring? The answer is that is invertible

if and only if detA is a unit in R. In order to show this, we need Cramer’s Rule, which is an important idea

even over fields. Let V = R

^{n}where R is a commutative ring, and let f : V → V be invertible. Cramer’s

rule is a technique to find f

^{ -1}(u) of a vector u ∈ V if f is invertible.

3. Explain how the problem of finding f

^{ -1}(u) is related to solving n linear equations in n unknowns:

Conclude that Cramer’s rule can be understood as a technique for solving linear equations.

4. Let and suppose f is invertible. Show that if then .

Show that and prove the following

**Theorem (Cramer’s Rule).**Let V = R

^{n}where R is a commutative ring, and let : V

^{n}→ R be the

normalized alternating n-linear functional for V . Assume that f : V → V is linear and invertible with

matrix A. Then if u ∈ V , the preimage is given by the formula

where is the jth column of A and u is put in the ith slot.

5. Suppose that . (Assume R = Q if you wish). Use

Cramer’s rule to find a vector v such that f(v) = (0, 2, 1). (Optional: now find it with row operations ).

6. A very important special case is computing since it gives the matrix for f

^{ -1}. Prove the following.

Use the theorem to find the matrix for f

^{ -1}where .

Assume R = Q if you wish. (Optional: now find it using row reduction ).

**Let V = R**

Theorem ( Inverse formula ).

Theorem ( Inverse formula ).

^{n}where R is a commutative ring. Also, assume that : V

^{n}→ R

is the normalized alternating n-linear functional for V . Finally assume that f : V → V is linear and

invertible with matrix A and inverse matrix B. Then is given by the formula

where is the jth column of A and is put in the ith slot. Also is the matrix for f

^{ -1}: V → V

(so B = A

^{ -1}). Thus A

^{ -1}can be computed with determinants.

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