Please Show in details Suppose you have a homogeneous system of linear equations

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Please Show in details

Suppose you have a homogeneous system of linear equations. How many solutions can it have? Suppose you have an inhomogeneous system of linear equations. How many solutions can it have? Suppose you have a linear transformation F: R^2 rightarrow R^2 and three distinct non-zero vectors vector u, vector v, vector w such that F(vector u) = F(vector v) and F(vector v) notequalto F(vector w). What are the possible ranks of F? Give an example of a matrix with each possible rank. Suppose that you have a 3 times 3 matrix A with nullity 2. Explain why there is some vector vector w such that ker(A) = {vector v elementof R^3: vector w middot vector v = 0}. Suppose that you have two independent vectors vector v and vector w in R^3 and two distinct real numbers b, c. Describe the "shape" and "position" of the set S = {u elementof R^3: u middot v = b, u middot w = c}. Consider the matrix A = (3 0 0 1 3 0 0 1 3) in the standard basis. Prove that there is no basis vector b_1, vector b_2, vector b_3 in which A would be written as a diagonal matrix D = (d_1 0 0 0 d_2 0 0 0 d_3).

Explanation / Answer

7.1 - A homogeneous system of equations can have 1 or infinte solutions.

equation will be AX = 0 type, So X = 0 is atleast one solution.

7.2 - A inhomogeneous equation can have No solution aur unique solution.

7.4 - Due to Rank-Nullity theorem, Rank + Nullity = 3, here nullity = 2, So Rank = 1. So Ker(A) will never be empty.

7.6 - Matrix A has a12 or a23 components, So it can't be expressed in the basis of diagonal matrix.

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