Find a basis for and the dimension of the following...

Find a basis for and the dimension of the following subspaces: (a) The space of solutions to the linear system $A \mathbf{x}=\mathbf{0}$, where $A=\left(\begin{array}{rrrr}1 & 2 & -1 & 1 \\ 3 & 0 & 2 & -1\end{array}\right)$. (b) The set of all quadratic polynomials $p(x)=a x^2+b x+c$ that satisfy $p(1)=0$. (c) The space of all solutions to the homogeneous ordinary differential equation $u^{\prime \prime \prime}-u^{\prime \prime}+4 u^{\prime}-4 u=0$.

Find a basis for and the dimension of the following subspaces: (a) The space of solutions to the linear system $A \mathbf{x}=\mathbf{0}$, where $A=\left(\begin{array}{rrrr}1 & 2 & -1 & 1 \\ 3 & 0 & 2 & -1\end{array}\right)$.
(b) The set of all quadratic polynomials $p(x)=a x^2+b x+c$ that satisfy $p(1)=0$. (c) The space of all solutions to the homogeneous ordinary differential equation $u^{\prime \prime \prime}-u^{\prime \prime}+4 u^{\prime}-4 u=0$.

Key Concepts

Subspaces of Polynomial Spaces

When working with polynomial spaces, conditions like evaluating to zero at a specific point reduce the overall dimension of the space by imposing linear constraints. This leads to a subspace where polynomials can be expressed in terms of a reduced set of parameters, and a basis can be constructed accordingly.

Row Reduction and Free Variables

Row reduction (or Gaussian elimination) is a systematic procedure used to simplify systems of linear equations, revealing the pivot positions and free variables. Free variables indicate the inherent degrees of freedom in the solution set, facilitating the extraction of a basis for the null space.

Homogeneous Linear Differential Equations and the Characteristic Equation

For homogeneous linear differential equations, solutions are often found by assuming exponential solutions which lead to the formation of a characteristic equation. The roots of the characteristic equation determine the fundamental set of solutions that span the solution space, with the order of the differential equation indicating its dimension.

Null Space (Kernel) of a Matrix

The null space of a matrix is the set of all vectors x that satisfy Ax = 0. It represents the solutions to a homogeneous system, and finding its basis typically involves performing row reduction to identify free variables and express the solution as linear combinations of parameter vectors.

Basis and Dimension

A basis of a subspace is a collection of linearly independent vectors that spans the space, and the dimension is defined as the number of vectors in the basis. These concepts are fundamental in linear algebra for describing the structure and degrees of freedom of various subspaces.

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