## How do you know if Ax B has a solution?

**Ax** = **b has a solution if** and only **if b** is a linear combination of the columns of A. Theorem 4 is very important, it tells us that the following statements are either all true or all false, for any m × n matrix A: (a) For every **b**, the equation **Ax** = **b has a solution**.

## Does the equation Ax B have a solution for each B in R4?

Since the matrix doesn’t **have** pivots in every row, it follows that the system **Ax** = **b** doesn’t **have a solution** for every **b** ∈ **R4**.

## Does the equation Ax B have at least one solution for every possible b?

only the trivial **solution** (because **every** column of A **has** a pivot position) and the **equation Ax** **b does have at least one solution for every possible b** (because **every** row of A **has** a pivot position). In fact, since **every** column of A **has** a pivot position, the **equation Ax** **b has** exactly **one solution for every possible b**.

## How many solutions does the equation Ax B have explain your answer?

The **equation Ax** = **b has an infinite** number of **solutions**. The **equation Ax** = 0 **has** a non-zero **solution**. Null(A) = {0}. A non–zero linear combination of the columns of A is equal to 0.

## What is the solution of Ax B?

One way to find a particular **solution** to the equation **Ax** = **b** is to set all free variables to zero, then solve for the pivot variables. The general **solution** to **Ax** = **b** is given by xcomplete = xp + xn, where xn is a generic vector in the nullspace.

## Do columns B span R4?

18 By Theorem 4, the **columns of B span R4** if and only if **B** has a pivot in every row. Therefore, Theorem 4 says that the **columns of B do** NOT **span R4**. Further, using Theorem 4, since 4(c) is false, 4(a) is false as well, so Bx = y **does** not have a solution for each y in **R4**.

## Is ax b consistent for all B?

The equation **Ax** = **b** is **consistent** if the augmented matrix [A **b**] has a pivot position in **every** row. Answer: False. The system is inconsistent if [A **b**] has a pivot in the last (“**b**“) column. The system is **consistent** if the matrix A has a pivot in **every** row.

## What is Ax B when does Ax B has a unique solution?

The system **AX** = **B has a unique solution** provided dim(N(A)) = 0. Since, by the rank theorem, rank(A) + dim(N(A)) = n (recall that n is the number of columns of A), the system **AX** = **B has a unique solution** if and only if rank(A) = n. A linear system of the form **AX** = 0 is said to be homogeneous.

## What kind of equation is Ax B 0?

The standard or ideal form of a linear **equation** with one variable is **ax** + **b** = , where a and **b** are constants, x is the variable, and a is not equal to . You can solve the **equation** for x to get x = − **b**/a. (See Solving a Linear **Equation** with One Variable for more information.)

## How do you write a parametric solution?

If there are m free variables in the homogeneous equation, the **solution** set can be expressed as the span of m vectors: x = s1v1 + s2v2 + ··· + smvm. This is called a **parametric** equation or a **parametric** vector form of the **solution**. A common **parametric** vector form uses the free variables as the parameters s1 through sm.

## How do you tell if a system of equations has no solution or infinitely many?

A **system** of linear **equations has** one **solution when** the graphs intersect at a point. **No solution**. A **system** of linear **equations has no solution when** the graphs are parallel. **Infinite solutions**.

## Is Ax BA a vector equation?

The **equation Ax**=b is referred to as a **vector equation**. The **equation Ax**=b has the same solution set as the **equation** x(1) a(1) + x(2) a(2) + + x(n) a(n) = b. The **equation Ax**=b is consistent if the augmented matrix [ A b ] has a pivot position in every row.