Question

We often write vectors in ${\mathbb{R}}^n$ as rows. If $A$ is an $m\times n$ matrix, show that, for each invertible $m\times m$ matrix $U,$ null $(A)=\mathrm{null}(UA)$.

Short answer

The null spaces of $A$ and $UA$ are equal.

Step-by-step solution

Define the Null Space

The null space of a matrix $A$, denoted as $\mathrm{null}(A)$, is the set of all vectors $\mathbf{x}$ in ${\mathbb{R}}^n$ such that $A\mathbf{x}=\mathbf{0}$. We need to prove that null $(A)=\mathrm{null}(UA)$ when $U$ is an invertible $m\times m$ matrix.

Assume a Vector in Null Space of A

Let $\mathbf{x}$ be any vector such that $A\mathbf{x}=\mathbf{0}$. This means $\mathbf{x}\in \mathrm{null}(A)$.

Show Inclusion $\mathrm{null}(A)\subseteq \mathrm{null}(UA)$

Starting from $A\mathbf{x}=\mathbf{0}$, if you left-multiply both sides by $U$, an invertible matrix, you get $U(A\mathbf{x})=U\mathbf{0}=\mathbf{0}$. Hence, $UA\mathbf{x}=\mathbf{0}$, which shows that $\mathbf{x}\in \mathrm{null}(UA)$. Therefore, every vector in $\mathrm{null}(A)$ is also in $\mathrm{null}(UA)$.

Show Inclusion $\mathrm{null}(UA)\subseteq \mathrm{null}(A)$

Now let $\mathbf{x}$ be any vector such that $UA\mathbf{x}=\mathbf{0}$. Since $U$ is invertible, there exists a matrix $U^{-1}$ such that $U^{-1}U=I$, the identity matrix. By left-multiplying by $U^{-1}$, we get $U^{-1}UA\mathbf{x}=U^{-1}\mathbf{0}=\mathbf{0}$. This simplifies to $A\mathbf{x}=\mathbf{0}$, which means $\mathbf{x}\in \mathrm{null}(A)$. Therefore, every vector in $\mathrm{null}(UA)$ is also in $\mathrm{null}(A)$.

Conclude Null Spaces Equality

Since we have shown $\mathrm{null}(A)\subseteq \mathrm{null}(UA)$ and $\mathrm{null}(UA)\subseteq \mathrm{null}(A)$, it's proven that $\mathrm{null}(A)=\mathrm{null}(UA)$.

Key concepts

Invertible Matrix

An invertible matrix is a special kind of matrix that has a unique property. If a matrix is invertible, it means there exists another matrix that can "undo" the effect of multiplying by the original matrix. In mathematical terms, for an invertible matrix $U$, there is a matrix $U^{-1}$ such that $U{U}^{-1}=I$, where $I$ is the identity matrix. The identity matrix acts like the number 1 in multiplication; anything multiplied by the identity matrix remains unchanged.

Invertibility is a key concept in linear algebra because it guarantees that linear transformations represented by these matrices are reversible. Not all matrices are invertible, though. A matrix must be square (same number of rows and columns) and have a non-zero determinant to ensure invertibility. If a matrix is not invertible, it is called singular.

Understanding invertible matrices is crucial because they enable us to solve systems of linear equations, find the null space of a matrix, and perform various calculations in linear algebra tasks.

Matrix Multiplication

Matrix multiplication is an essential operation in linear algebra, often used to combine transformations or datasets. It involves rows and columns interacting in a specific way. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.

Each element in the resulting matrix is calculated by taking a row from the first matrix and a column from the second matrix, multiplying corresponding entries, and adding them up. This process forms the foundation for understanding many concepts in linear transformations and systems of equations.

For example, if you have a matrix $A$ with dimensions $m\times n$ and another matrix $B$ with dimensions $n\times p$, their product $AB$ will be an $m\times p$ matrix. It's important to follow the correct order, as matrix multiplication is not commutative: $ABeqBA$ in general.

This non-commutativity property plays a critical role in the behavior of null spaces when multiplied by different matrices, such as when proving that $\mathrm{null}(A)=\mathrm{null}(UA)$ with $U$ being invertible.

Vector Spaces

Vector spaces are mathematical environments where vectors can be added together and multiplied by scalars to produce new vectors. They are foundational to linear algebra. A vector space must satisfy certain rules, such as closure under addition and scalar multiplication, as well as containing a zero vector.

When studying linear transformations and null spaces, we often use vector spaces to understand the structure and dimensionality of these transformations. The null space of a matrix, for example, is a vector space consisting of all vectors that the matrix sends to the zero vector. In other words, for a matrix $A$, any vector $\mathbf{x}$ fulfilling $A\mathbf{x}=\mathbf{0}$ resides in the null space of $A$.

These concepts help describe and analyze different subspaces within a given vector space, offering insights into potential solutions for linear equations and helping articulate geometric interpretations of vector operations.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, matrices, and linear transformations. It provides the language and tools to analyze linear equations, spaces, and systems, which are used widely in science, engineering, and analytics.

Key elements of linear algebra include vector spaces, linear mappings, and matrix representations. Solutions to systems of linear equations, eigenvalues, and eigenvectors are also crucial components. Understanding these concepts allows us to manipulate and solve complex problems.

For instance, linear algebra helps us comprehend the significance of null spaces, where we seek vectors that, when multiplied by a matrix, yield the zero vector. Such studies pave the way for grasping how transformations affect data, the conditions for system solvability, and how we progress to advanced topics like singular value decomposition and principal component analysis.

Linear algebra is not only theoretical but also practical. It underpins many algorithms in computer science, optimization problems, and statistical analyses, making it essential for students to master its concepts thoroughly.