Intereting Posts

The meaning of implication in logic
Find all solutions of ${\frac {1} {x} } + {\frac {1} {y} } +{\frac {1} {z}}=1$, where $x$, $y$ and $z$ are positive integers
A maximal ideal among those avoiding a multiplicative set is prime
Badly explained solution
Adding or Multiplying Transcendentals
How many solution of a equations?
There does not exist a polynomial $p(x)$ with integer coefficients which gives a prime number $\forall x\in \mathbb{Z}$
surface area of torus of revolution
Deck transformation acting properly discontinuously assumed covering space is path-connected
Number of divisiors of $n$ less than $m$
Cubes of binomial coefficients $\sum_{n=0}^{\infty}{{2n\choose n}^3\over 2^{6n}}={\pi\over \Gamma^4\left({3\over 4}\right)}$
Number of irreducible polynomials with degree $6$ in $\mathbb{F}_2$
Polynomial Interpolation and Security
Double Integral of xy
Correlation matrix from Covariance matrix

What I did:

I put this into reduced row echelon form:

- Example of a linear operator on some vector space with more than one right inverse.
- How to find perpendicular vector to another vector?
- Why is every irreducible matrix with period 1 primitive?
- Prove that every real vector space has infinitely many vectors
- Determinant value of a square matrix whose each entry is the g.c.d. of row and column position
- Determinant of matrix composition

$$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$

It is clear that the $r(M)=2$, because there are two independent rows.

Now for the null space, I wrote down the equations from the reduced row echelon form:

$$x-2y+2z+4t=0$$

$$z+t=0$$

I can’t seem to write $x$ and $y$ separately in terms of $z$ and $t$. Any hints?

- Exponential function and matrices
- Determinant-like expression for non-square matrices
- Prove that if $(v_1,\ldots,v_n)$ spans $V$, then so does the list $(v_1-v_2,v_2-v_3,\ldots,v_{n-1}-v_n,v_n).$
- Extending a linear map
- tangent space of manifold and Kernel
- Jordan decomposition of an endomorphism with minimal polynomial
- Why does Friedberg say that the role of the determinant is less central than in former times?
- Proving Distributivity of Matrix Multiplication
- Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
- Matrix with non-negative eigenvalues (and additional assumption)

Write the system

$$\left\{\begin{array}{rcl} x-2y+2z+4t& =& 0 \\ z+t & = & 0\end{array}\right.$$

as

$$\left\{\begin{array}{rcl} x+2z& =& 2y-4t \\ z & = & -t\end{array}\right.$$ and solve it. You get,

$$\left\{\begin{array}{rcl} x& =& 2y-2t \\ z & = & -t\end{array}\right.$$ That is, $$(2y-2t,y,-t,t)$$ is an element of the null space for any $y,t.$ Now, look for two linearly independent vectors.

(Note that the kernel has dimension $2.$ So the system has infinitely many solutions that have to depend on two parameters.)

Do also backwards elimination:

$$

\begin{bmatrix}

1 & -2 & 2 & 4 \\

0 & 0 & 1 & 1 \\

0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0

\end{bmatrix}

\to

\begin{bmatrix}

1 & -2 & 0 & 2 \\

0 & 0 & 1 & 1 \\

0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0

\end{bmatrix}

$$

Now your equations read

$$

\begin{cases}

x_1=2x_2-2x_4\\

x_3=-x_4

\end{cases}

$$

You get two linearly independent vectors in the null space by setting $x_2=1, x_4=0$ and $x_2=0,x_4=1$, so the basis is given by the two linearly independent vectors

$$

\begin{bmatrix}

2\\

1\\

0\\

0\\

\end{bmatrix}

\qquad

\begin{bmatrix}

-2\\

0\\

-1\\

1

\end{bmatrix}

$$

The first corresponds to $x_2=1$ and $x_4=0$, the second to $x_2=0$ and $x_4=1$.

Just to make the answer a bit more algorithmic: a “pivot entry” is an entry which is the first non-zero entry in its row. A “pivot column” is a column containing a pivot entry. In your matrix, columns 1 and 3 are pivot columns.

Name the variables after the columns as you did (so, $x, y, z, w$). Then the “free variables” are the ones that don’t occur in pivot columns. In your case, these are $y$ and $w$. The remaining rows of the matrix express the bound variables in terms of the free variables.

First case: $z=t=0$, you obtain $x-2y=0$. One of solutions is the vector $(2,1,0,0)$.

Second case: $z=-t=1$, which gives you the equation $x-2y=2$, which gives you, for example, $(2,0,1,-1)$.

Write $z = -t$ and put it in your first equation. You shall get $x = 2y + 6t$, after simplification. Write $$(x,y,z,t)’ = y(2 , 1 , 0 , 0)’ + t(6, 0, -1, 1)’$$. Now consider $y$ and $t$ in some field. This shall give you all the solutions.

- The cardinality of the set of all linear order types over $\omega$ is $2^{\aleph_0}+\aleph_1$ in ZF+AD?
- Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular
- Curvature of the metric $ds^2=y^2dx^2+x^2dy^2$
- Suppose that a sequence is Cesaro summable. Prove…
- Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1.
- On the weak closure
- epsilon-dense property
- St. Petersburg Paradox
- The Relationship Between Cohomological Dimension and Support
- Congruence Class $_5$ (Equivalence class of n wrt congruence mod 5) when n = $-3$, 2, 3, 6
- Why is the determinant defined in terms of permutations?
- A limit wrong using Wolfram Alpha
- Determining stability of the differential equation
- How to construct $\{\{\{…\}\}\}$ in ZF without axiom of foundation
- Generalization of the Jordan form for infinite matrices