Find a basis for the orthogonal complement of the following...

Find a basis for the orthogonal complement of the following subspaces of $\mathbb{R}^3$ : (a) the plane $3 x+4 y-5 z=0 ;(b)$ the line in the direction $(-2,1,3)^T ;(c)$ the image of the $\operatorname{matrix}\left(\begin{array}{rrrr}1 & 2 & -1 & 3 \\ -2 & 0 & 2 & 1 \\ -1 & 2 & 1 & 4\end{array}\right)$; (d) the cokernel of the same matrix.

Key Concepts

Image of a Matrix

The image (or column space) of a matrix is the set of all vectors that can be obtained as linear combinations of the matrix's columns. It represents the output of a linear transformation defined by the matrix and is crucial for assessing the solvability of linear systems and for concepts like rank.

Subspace

A subspace is a subset of a vector space that is closed under both vector addition and scalar multiplication. It is important because it represents a smaller, yet fully valid, vector space where many properties of the larger space can be studied in a simpler context.

Cokernel

The cokernel of a linear transformation, or matrix, is defined as the quotient of the codomain by the image of the transformation. This concept, which is related to the rank and nullity of the matrix, helps in understanding the structure and deficiencies of a transformation in terms of which vectors in the codomain are not 'covered' by the map.

Orthogonal Complement

In an inner product space, the orthogonal complement of a subspace consists of all vectors that are perpendicular to every vector in the given subspace. This concept is essential for decomposing spaces into mutually perpendicular parts and plays a central role in projections, least squares problems, and many areas of linear algebra.

Basis

A basis is a set of linearly independent vectors that span a vector space or subspace. Every element of the space can be expressed uniquely as a linear combination of the basis vectors, making the concept key for understanding the structure and dimension of vector spaces.

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