Linear Algebra (Self Doubt)

Question

Let $A$ be a $n \times n$ square matrix whose all columns are independent. Is $Ax = b$ always solvable?

Actually, I know that $Ax= b$ is solvable if $b$ is in the column space of $A$. However, I am not sure if it is solvable for all values of $b$.

Comments

Let $A = \begin{bmatrix} col_{1} & col_{2} & col_{3} \end{bmatrix}$ be any arbitrary $3 \times 3$ matrix.

Here, $col_{1}, col_{2}, col_{3}$ are the column vectors of A, each of dimension $3 \times 1$.

Also, let $x =\begin{bmatrix} x_{1}\\ x_{2}\\ x_{3} \end{bmatrix}$.

$ \therefore Ax = b \Rightarrow col_{1}.x_{1} + col_{2}.x_{2} + col_{3}.x_{3} = b$, i.e., a linear combination of the column vectors of $A$.

Therefore, finding the solution of the equation $Ax = b$ basically reduces to finding a linear combination of column vectors of matrix $A$ that result in vector $b$. Such a combination can only be found if vector $b$ lies in the column space of matrix $A$, otherwise not.

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$\begin{bmatrix} 1 & 2 & 3\\ 4& 5 &6 \\ 7& 8 & 10 \end{bmatrix}$ $\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$ $=$ $\begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}$

This has a solution of $\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$ $=$ $\begin{bmatrix} -(2/3)\\ -(2/3)\\ 1 \end{bmatrix}$

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yes right.

@ankitgupta.1729

u r not correct here

Answer 1

Few Inferences first :

• If the Rank of the Augmented Matrix $[A:b]$ $=$ Rank $[A]$ =$N$(Order of the matrix A) then it is solvable and can be solved by reducing by using Gaussian Elimination Method, and it has Unique Solution.
• If the  Rank of the Augmented Matrix $[A:b]$  $=$ Rank $[A]$$ < N$ then it is solvable and has Infinitely many solutions.
• If the  Rank of the Augmented Matrix $[A:b]$  $\neq$ Rank $[A]$ then it is not solvable.

Now as per your question, if A is a vector whose all columns are linearly independent then in that case it has a rank = $n$ and whatever may be $b$ the augmented matrix $[A:b]$ will always have a rank = $n$ hence it is always solvable and will have unique solution.

For more details you can refer to the lectures by Gilbert Strang or his book on linear algebra.

Comments

Thanks. I was doing Gaussian Elimination but did some mistake.