Linear Dependence and Basis in Vector Spaces | Matrix Determinant Properties

[alexngo]

If two non-zero geometric vectors are parallel then they are linearly dependent.

For all n x n matrices A,B and C, we have (A - B)C = CA - CB

Let V be a vector space. If S is a set of linearly independent vectors in V such that S spans V,then S is a basis for V.

For all n x n matrices A and B, we have det (A + B) = det (A) + det (B)

If A is a matrix with det(A) = 0, then there are no solution to the equation Ax = b for any column vector b where b is not equal to 0

Row operations on an n x n matrix A have no effect on the determinant of A

For all n x n matrices A, we have det(A^t) = det(A)

 

[CompuChip]

What do you think?
1. For the first one, you could start by looking up the definition of "linearly dependent", for example.
2. matrix multiplication is associative.
3. What is a basis?
4. look up the properties of the determinant.
5. You can solve Ax = b by multiplying by the inverse of A from the left.
6 and 7: look into the properties of the determinant again.

 

[CaffeineJunky]

1. When is a set of two vectors linearly dependent?

2. It is certainly true that (A - B)C = AC - BC, since matrix multiplication distributes. However, is it always true that AC - BC = CA - CB?

3. What is the definition of a basis?

4. You can easily construct a counterexample to show that this is false. (Use matrices that are almost all zero and the identity matrix.)

5. There's a solution if and only if a = A^{-1}b, which is true as long as A^{-1} exists. Does det(A) = 0 say anything about the existence of A^{-1}?

6. How does each of the following row operations affect the determinant of a matrix: (1.) Multiplying one row by a scalar multiple of another row, (2.) Interchanging two rows, (3.) Adding a scalar multiple of one row to another row?

7. This is almost trivial.

 

[jacobrhcp]

it seems as though you can solve all these yourself within 15 minutes probably! You certainly come across bright enough. Just look up the words for things you don't know the definition of, I'm sure they are listed in the register of your book or even wikipedia or google. Then if you don't understand, come back.

for 5, this is a theorem I'm sure your book states, otherwise think of when a matrix can be swept to the identity by row operations. Then use det(I) is not zero (you might want to use the result of 6 here though)

6, try one or two simple 2x2 or 3x3 matrices and I'm very sure you will have a suspicion about the determinant rules, which is easy to explain. Try it, you'll succeed.