Material to know

1. Coordinate with respect to a basis

2. Matrix of change of a basis

3. matrix of a linear transformation with respect to the given basis

4. Two matrices are similar

5. dimension of a liner space, subspace

6. Linear transformation:

   1. what is linear transformation ?

   2. Nullity

   3. Rank

   4. Kernel

   5. Image

   6. Rank-Nullity Theorem

   7. isomorphism

7. Comfortable with linear spaces

   1. spaces of polynomials, P_1, P_2, P_3 and its subspaces

   2. Spaces of matrices such as R^2x2 and Subspaces of R^2x2 such as

      1. the set of all upper triangular matrices,

      2. diagonal matrices,

      3. the set of matrices whose determinant is 0,

   3. basis, coordinate of a vector with respect to the given basis

   4. linear transformation and its matrix

8. Properties of determinant

   1. Linearity in its columns

   2. Determinant of the identity matrix I_n is 1.

   3. Change in determinant with regards to elementary row operations, and column operations

   4. Cramer's Rule for a solution for Ax=b

   5. Volume of a parallelepiped

   6. For nxn matrix A, the following are equivalent

      1. Determinant of A is non-zero,

      2. RREF is the identity,

      3. non-singular matrix,

      4. solution to Ax=0 has a unique solution(trivial), Ker(A)={0}

      5. Columns of A are linearly independent

      6. Rows of A are linearly independent

Sample Questions

1. Textbook Section 3.4: #28, 44, 16, 56

2. Section 4.1:4, 6, 16,18

   1. For #18 Find at least two basis for P_3

3. Section 4.2: #6,52

4. Section 4.3: 1 , 6, 20

5. Section 6.1: 39,

6. Section 6.2: #47 If the answer is yes, is T determinant map?

   What condition does it violate to be a determinant?

   #59. Hint: Consider the case when det =0.

7. Section 6.3: 224,

8. True False Chapter Tests

9. Web6, Web7, Web8

10. Consider P_2 the space of polynomials with degree <=2. Consider the linear map T:P_2 ----> P_2 defined by T(f(t))=f(t+1). Find the matrix of T with respect to the basis of B=(1,1+t,1+t+t²)

11. Solution will be out soon.