Math 2890 Quiz 2 Show All Work Clearly And In Detail

Math 2890 Quiz 2show All Work Clearly And In Detail

Construct a 6 x 6 non-zero matrix A and a vector b in ℝ⁶ such that b is not in the set spanned by the columns of A.

Given the requirement, we need to create a matrix A with six columns (each in ℝ⁶), all non-zero vectors, but such that there exists at least one vector b in ℝ⁶ that cannot be expressed as a linear combination of the columns of A. This implies that the columns of A are linearly dependent, but do not span the entire space ℝ⁶. Therefore, the rank of A should be less than 6, and b should be outside the span of the columns of A.

Constructing the matrix A

Let us choose A to have four non-zero vectors in its columns, deliberately making the columns linearly dependent, but not enough to span all of ℝ⁶. For example:

In matrix form,

A =

\[

\begin{bmatrix}

1 & 0 & 0 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 & 0 & 0 \\

0 & 0 & 1 & 0 & 0 & 0 \\

0 & 0 & 0 & 1 & 0 & 0 \\

0 & 0 & 0 & 0 & 1 & 0 \\

0 & 0 & 0 & 0 & 0 & 0

\end{bmatrix}

\]

Note that the sixth row is all zeros, indicating the rank of A is 5, hence the span of its columns is a 5-dimensional subspace of ℝ⁶.

Choosing vector b not in span of columns of A

Any vector b in ℝ⁶ that has a non-zero sixth component will not be in the span of the columns of A because the columns span only vectors with zeroes in the sixth component. For instance,

b = (0, 0, 0, 0, 0, 1)

which is not in the span, since the last component cannot be generated by linear combinations of the columns with zero in the sixth position. Therefore, this b is outside the span of the columns of A.

Summary of part 1

We constructed a 6x6 matrix A with rank 5, whose columns are linearly dependent, and selected a vector b with a non-zero sixth component that is specifically outside the span of A's columns. This satisfies the condition that b is not in the span of the columns of A.

Second Part: Span of Columns in Reduced Echelon Form

The second question involves a matrix in reduced echelon form (REF) that spans ℝ⁴. The explicit matrix in question is:

[24  -14  22  -12]
[-18  8  -16  10]
[-6  -12  22  -14]
[0  -18  8  -12]
[20  -4  24  ]]

Assuming this matrix is in REF (or RREF) form, the columns that are pivotal (leading columns) span ℝ⁴. To find a column that can be deleted while still maintaining the span of ℝ⁴, we observe which columns correspond to leading variables.

Typically, in RREF, the pivot columns are the first four, indicating that the first four columns are essential to span ℝ⁴. Therefore, deleting a non-pivot column—one that does not contain a leading 1—will not affect the span.

Identifying removable columns

Suppose, in the augmented matrix, the fourth column is a free variable and the first three, along with the second, are pivots. Therefore, the fourth column may be deleted without losing the ability to span ℝ⁴.

Thus, removing Column 4 (assuming it is free) results in the remaining columns still spanning ℝ⁴.

Third Part: Parametric Solutions of Systems of Equations and Geometric Comparison

Given the two systems:

System 1:

a) \( x + y + z = 1 \)

b) \( 2x - y + 2z = 2 \)

c) \( -x + 3y - z = 0 \)

and system 2: (not explicitly specified here, but assumed to be a similar system with different constants)

For system 1, we proceed to write the augmented matrix and reduce it to parametric form. The solution involves expressing free variables and describing the solution set as a line or plane in ℝ³.

Assuming the second system has inconsistent equations or different constraints, its solution set may be empty, a point, a line, or a plane, depending on the rank and consistency.

Geometrically, the solutions of the first system could form a line or a plane in 3D space, depending on the number of free variables after reduction. The second system’s solution set could be disjoint, contained within, or intersecting the first, which can be analyzed based on their parametric forms and ranks.

Conclusion

The parametric form reveals the geometric structure: the set of points satisfying the equations forms a geometrical object in space, offering insight into possible intersections, containment, or emptiness of the solution sets of the two systems.

References

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