Question

Ive an orthonormal basis for the row space - Give an orthonormal basis for the column space - Justify that the two bases are orthonormal - Express every column vector of the matrix as a linear combination of the column space basis.1. ( 3 −3 0

−3 −2 −4
0 −4 −2 )
2. ( −2 −8 −3 10
3 7 2 −10
−4 −10 −3 14 )

Answer 1

An orthonormal basis for the row and column space of a matrix is determined using the Gram-Schmidt process and normalization, ensuring vectors are orthogonal and unit length. Each matrix column vector can be expressed as a linear combination of the column space basis.

Step-by-step explanation:

To find an orthonormal basis for the row space and column space of a given matrix, we need to implement the Gram-Schmidt process followed by normalization. The orthonormal basis for the row space consists of vectors that are orthogonal to each other and have unit length. The same applies to the column space. These bases are orthonormal because their vectors are mutually orthogonal and all have a norm of 1. Moreover, unit vectors in orthogonal coordinate systems, as in the Cartesian coordinate system, are inherently orthonormal. They satisfy conditions such as cos 90° = 1⋅ 1⋅ 0 = 0, indicating that the dot product between distinct unit vectors is zero, affirming their orthogonality.

Each column vector of the matrix can be expressed as a linear combination of the basis vectors of the column space. To do this, we project each column vector onto each of the basis vectors and sum these projections.