1. the matrix below is nonsingular. what can you now say about its columns? r-3 0 11 a 1 2 1 2. write the vector was a linear combination of the columns of the 15 matrix a above. how many ways are there to answer this question? 3. why is an orthonormal basis desirable?

To answer the multi-part question clearly:

1. If the matrix is nonsingular (invertible), its columns are linearly independent and therefore form a basis for the space (e.g., Rn\mathbb{R}^nRn) that the matrix operates on. This means no column can be written as a linear combination of the others.
2. Any vector v\mathbf{v}v that is a linear combination of the columns of matrix AAA means there exist coefficients c1,c2,…,cnc_1,c_2,\ldots,c_nc1​,c2​,…,cn​ such that

v=c1a1+c2a2+⋯+cnan\mathbf{v}=c_1\mathbf{a}_1+c_2\mathbf{a}_2+\cdots +c_n\mathbf{a}_nv=c1​a1​+c2​a2​+⋯+cn​an​

where ai\mathbf{a}_iai​ are the columns of AAA. Since AAA is nonsingular (and column vectors form a basis), this representation is unique, so there is exactly one way to express v\mathbf{v}v as a linear combination of the columns.

3. An orthonormal basis is desirable because its vectors are not only linearly independent but also orthogonal and each has unit length. This simplifies many operations:
   • Coordinates of any vector relative to an orthonormal basis are easy to find using inner products.
   • Linear transformations, projections, and decompositions become computationally simpler and numerically more stable.
   • Orthonormal bases facilitate easier representation and manipulation of vectors and functions in various applications such as signal processing and numerical analysis.

References:

• The nonsingularity implies linear independence of columns and unique representation of vectors.

• Vector as linear combination is uniquely determined in case of nonsingular matrices.

• Reasons for desirability of orthonormal basis include ease of coordinate representation and computational advantages.