Let A = I. Find A Basis For The Column Space Of A. Ii. Expan

Let A= i. Find a basis for the column space of A. ii. Expand your basis for the column space to a basis of the entire range of A. iii. Find a basis for the null space (or kernel) of A. iv. Find a basis for the domain of A like that given in Theorem 6.8. (Theorem 6.8: domain, image, and null spaces)( in applied linear algebra: page252 ) v.

Given the complex and somewhat unclear data, the primary focus is on analyzing a linear transformation represented by matrix A, with the goal of identifying bases for its column space, null space, and domain, in accordance with linear algebra principles outlined in Theorem 6.8. This process involves performing a systematic analysis using Gaussian elimination, matrix rank, and the properties of linear transformations, particularly the concepts of the column space (range), null space (kernel), and domain.

Analysis and Solution

1. Understanding the Matrix A

The data provided appears to contain numerical sequences intermingled with text and unclear symbols. For the purpose of a rigorous mathematical solution within an applied linear algebra context, it is necessary to assume that matrix A is explicitly given, or that the relevant data can be formatted into a matrix. Due to the information's ambiguity, a typical approach involves constructing a matrix that captures the numerical relationships indicated.

Suppose A is a matrix derived from the numeric sequences, with the key task being to analyze its column space, null space, and related subspaces. Given the translated data, assume A is a matrix with dimensions compatible with the listed vectors, such as a 3x3 or 4x4 matrix.

2. Finding a basis for the column space of A

The column space of A is spanned by the columns of A. To find a basis, we must identify the linearly independent columns.

- Perform Gaussian elimination to reduce A to its echelon form.

- Identify the pivot columns—the columns corresponding to leading entries.

- The pivot columns from the original matrix A form a basis for its column space.

Suppose, after elimination, columns 1, 2, and 4 contain pivots. The basis for the column space would then be these columns from A.

3. Expanding the basis to the entire range of A

The column space describes the range or image of the linear transformation; therefore, the basis obtained above from the pivot columns already spans the entire range of A. No further expansion is necessary unless the matrix contains additional vectors that can be included to span larger subspaces; however, typically, the basis for the column space is a basis for the entire range.

4. Finding a basis for the null space of A

The null space consists of all solutions to the homogeneous system Ax=0.

- Using the row echelon form of A, express the free variables.

- Assign parameters to the free variables.

- Express the solution vectors in parametric form.

- These vectors form a basis for the null space.

For example, if the reduced form shows variables y and z are free, the null space basis can be constructed by setting each free variable to 1 in turn and the others to 0, generating basis vectors.

5. Basis for the domain of A

The domain of A, for a matrix transformation, corresponds to the vector space of the variables, typically R^n, where n is the number of columns.

- The basis for the domain is commonly taken as the standard basis vectors in R^n.

- Alternatively, in the context of Theorem 6.8, the basis can be restricted to vectors not mapped to zero to reflect the image's basis.

Since the null space's basis vectors are independent, the domain basis can be associated with the standard basis, with the dimension equal to the number of columns in A.

6. Explicit formula for solutions to Ax=b

Given a specific vector b, the solutions x can be expressed as:

x = x_p + x_h,

where x_p is a particular solution to Ax=b, and x_h is an arbitrary vector in the null space (homogeneous solutions).

- Find a particular solution via substitution or matrix inverse (if invertible).

- The general solution combines particular and homogeneous solutions, with the free variables parameterizing the null space basis.

Conclusion

This analysis relies on standard linear algebra methods: Gaussian elimination, basis identification through pivot columns, null space parametrization, and understanding the relationships among the domain, range, and null space via Theorem 6.8. Due to the incomplete and complex data provided, the steps outlined serve as a framework; actual computations depend on the explicit entries of matrix A, which would be precisely determined in a concrete case.

References

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