linear alegrab

Assignment #1

1. Determine whether or not the following linear system of 4 equations in 5 variables x1,x2,x3,x4,x5 has a solution.

x1 + 2x2 + 3x3 + 4x4 + 5x5 = 6

-x1 + x2 - x3 + x4 - x5 = 1 x2 + x4 = 2

x1 + x3 + x5 = 1

If there is exactly one solution, solve for it. If there are infinitely many solutions, parameterize the solution set. If there are no solutions, say that there are no solutions.

2. [Consider the following linear system of 2 equations in 4 variables x , x , x , x .

x1 + 2x3 - 4x4 = 0

x2 + 3x4 = 0

The solution set of this linear system can be parametrized using two parameters t1,t2 and two vectors v1,v2 in the form

s(t1,t2) = t1v1 + t2v2.

Find the vectors v1,v2 given the information that

and .

3. Denote by v1, v2, v3, and v4 the following vectors of R3:

, and .

(a) Is the span Span{v , v , v , v } all of R3? Why or why not?

(b) Determine if the vectors v , v , v , and v form a linearly dependent or linearly independent set of vectors. Justify your answer.

4. Provide counterexamples to the following claims.

(a) Every linear system of 3 equations in 3 variables has a unique solution.

(b) There do not exist three vectors in R3 that are linearly independent.

(c) There are no linear transformations T : R3 ?R4 that are one-to-one. (d) If a linear transformation T : R3 ?R2, is onto then T is one-to-one.

5. Let T : R4 ?R3 be the linear transformation defined by the formula T(x) = Mx where the matrix M is defined as

and x is any vector of R4.

(a) Show that T is onto.

(b) Show that T is not one-to-one.

(c) Give a parametrization for the solution set of the linear system defined by the equation Mx = v where

?x1?

x = ??xx23??? and.

?

x4

2

Assignment #1

1. Determine whether or not the following linear system of 4 equations in 5 variables x1,x2,x3,x4,x5 has a solution.

x1 + 2x2 + 3x3 + 4x4 + 5x5 = 6

-x1 + x2 - x3 + x4 - x5 = 1 x2 + x4 = 2

x1 + x3 + x5 = 1

If there is exactly one solution, solve for it. If there are infinitely many solutions, parameterize the solution set. If there are no solutions, say that there are no solutions.

2. [Consider the following linear system of 2 equations in 4 variables x , x , x , x .

x1 + 2x3 - 4x4 = 0

x2 + 3x4 = 0

The solution set of this linear system can be parametrized using two parameters t1,t2 and two vectors v1,v2 in the form

s(t1,t2) = t1v1 + t2v2.

Find the vectors v1,v2 given the information that

and .

3. Denote by v1, v2, v3, and v4 the following vectors of R3:

, and .

(a) Is the span Span{v , v , v , v } all of R3? Why or why not?

(b) Determine if the vectors v , v , v , and v form a linearly dependent or linearly independent set of vectors. Justify your answer.

4. Provide counterexamples to the following claims.

(a) Every linear system of 3 equations in 3 variables has a unique solution.

(b) There do not exist three vectors in R3 that are linearly independent.

(c) There are no linear transformations T : R3 ?R4 that are one-to-one. (d) If a linear transformation T : R3 ?R2, is onto then T is one-to-one.

5. Let T : R4 ?R3 be the linear transformation defined by the formula T(x) = Mx where the matrix M is defined as

and x is any vector of R4.

(a) Show that T is onto.

(b) Show that T is not one-to-one.

(c) Give a parametrization for the solution set of the linear system defined by the equation Mx = v where

?x1?

x = ??xx23??? and.

?

x4

2

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