Find an explicit description of Nul A by listing vectors that span the null spac

Trent Carpenter

Trent Carpenter

Answered question

2021-09-18

I need to find a unique description of Nul A, namely by listing the vectors that measure the null space.
A=[154310121000000]

Answer & Explanation

Benedict

Benedict

Skilled2021-09-19Added 108 answers

First find the general solution of Ax=0 in terms of the free variables. Since
[A 0]=[154310012100000000]R1R15R2[106810012100000000]
the general solution is
x1=6x3+8x4x5
x2=2x3x4
with x3,x4 and x5 free. So
x=[x1x2x3x4x5]=x3[62100]+x4[81010]+x5[10001]
and a spanning set for Nul A is
{[62100],[81010],[10001]}
Result:
{[62100],[81010],[10001]}

 

Jazz Frenia

Jazz Frenia

Skilled2023-05-10Added 106 answers

To find a unique description of the null space of matrix A, we need to list the vectors that measure the null space. The matrix A is given by:
A=[154310121000000]
To determine the null space, we solve the equation A𝐱=0, where 𝐱 is a vector in the null space and 0 is the zero vector. Writing this in augmented form, we have:
[154310121000000][x1x2x3x4x5]=[000]
Simplifying the matrix equation, we get the following system of equations:
x1+5x24x33x4+x5=0x22x3+x4=00=0
To find the null space, we solve this system of equations. First, we can express x1, x2, and x5 in terms of the remaining variables:
x1=4x3+3x4x5x2=2x3x4
Now, we can write the general solution for the null space vector 𝐱 as follows:
𝐱=[4x3+3x4x52x3x4x3x4x5]
where x3, x4, and x5 are arbitrary constants.
Thus, the unique description of the null space of matrix A is given by the set of vectors 𝐱:
𝐱=[4x3+3x4x52x3x4x3x4x5]
where x3, x4, and x5 can be any real numbers.
Nick Camelot

Nick Camelot

Skilled2023-05-10Added 164 answers

The null space of a matrix A is the set of all solutions 𝐱 to the equation A𝐱=0, where 0 is the zero vector.
For the given matrix A=[154310121000000], we need to find the null space. Writing the equation in augmented form and reducing the matrix to echelon form, we get:
[154310121000000][106210121000000]
From this, we can see that the first and fourth columns of the matrix contain the leading variables x1 and x4, respectively. The second and third columns contain the free variables x2 and x3, respectively.
Using the leading and free variables, we can express the general solution for the null space vector 𝐱 as:
𝐱=[6x3+2x4x52x3x4x3x4x5]=x3[62100]+x4[21010]+x5[10001]
where x3, x4, and x5 are arbitrary constants.
Therefore, the null space of matrix A can be described as the span of the set of vectors:
{[62100],[21010],[10001]}
which consists of three linearly independent vectors, and thus forms a basis for the null space.
Mr Solver

Mr Solver

Skilled2023-05-10Added 147 answers

Answer:
[32100] and [21010]
Explanation:
To find a unique description of the null space of matrix A, we need to determine the vectors that measure the null space. Given the matrix:
A=[154310121000000]
We can find the null space of A by solving the equation A𝐱=0, where 𝐱 is a vector in the null space. Let's write the augmented matrix and perform row reduction to solve for 𝐱:
[154310012100000000]
Performing row operations, we can obtain:
[103210012100000000]
Now, we can express the solution in terms of free variables. Let x3=t and x4=s, where t and s are arbitrary parameters. Then, the solution vector 𝐱 is given by:
𝐱=[3t+2s2tsts0]
Thus, the null space of matrix A can be expressed as:
Null(A)={[3t+2s2tsts0]t,s}
This means that any vector in the null space of A can be written as a linear combination of the vectors [32100] and [21010].

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