Linked Questions

593
votes
13answers
112k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
15
votes
3answers
2k views

Basis for $\mathbb{Z}^2$

Let $x = (a, b), y = (c, d) \in \mathbb{Z}^2$. What is the condition on $a, b, c, d$ so that ${x, y}$ is a basis? My answer: $ad\neq bc$ and $gcd(a, c) = gcd(b, d) = 1$. The first condition ensures ...
14
votes
4answers
2k views

Best way of introducing determinants in a linear algebra course

What is the best way of introducing determinants in a linear algebra course? I want to give real life examples of where the determinant is applied. It should have a real impact.
3
votes
8answers
1k views

Intuition behind determinant of a matrix with $2$ equal rows [closed]

In my linear algebra course, we have just proved that if a matrix $A$ contains $2$ equal rows, then $\det(A)=0$. I understand how the proof works, but could somebody offer a more intuitive ...
3
votes
4answers
309 views

Proof the area of a given triangle with coordinates is half determinant

I was given a problem, tried to solved it but couldn't get to a solution. It goes like that: There's a triangle ABC with area S. $$ \vec{AB} = (a,b) $$ $$ \vec{AC} = (c,d) $$ Prove that $$ S = \...
7
votes
2answers
638 views

Geometric interpretation of Laplace's formula for determinants

Coming from the geometric point of view, the determinant of an $n \times n$-Matrix computes the volume of an parallelepiped spanned by the columns of the matrix. In context of this question, let the ...
1
vote
2answers
3k views

Proving formula to find area of triangle in coordinate geometry.

Given 3 points, $A$, $B$ and $C$ in anti clockwise order, I have to find the area of the $\triangle ABC$. The formula is area $=\frac{1}{2}(A_x*B_y+B_x*C_y+C_x*A_y-A_y*B_x-B_y*C_x-C_y*A_x)$. Here $...
5
votes
1answer
933 views

Cramer's rule: Geometric Interpretation

I have a question concerning Cramer's rule: Let $A$ be a matrix and $A \cdot \vec x = \vec b$ a lineare equation. $A_i$ is the matrix $A$ where the i'th column is replaced by $\vec b$ if $det(A) \...
0
votes
1answer
584 views

cross product of sides of a triangle

i am given with a triangle and and i read here that i can cross product of the two sides of a triangle gives it orientation. what does cross product of sides of a triangle mean. here is the formula: ...
1
vote
0answers
539 views

Area of Parallelogram + Cross Product

This question is nearly identical - I would have commented there, but I suppose the post is too old. I would like to extend that scenario slightly. Using Tpofofn's diagram - vector (a,b) is clockwise ...
4
votes
2answers
104 views

How do I demonstrate the geometric meaning of a $2\times2$ matrix?

Given the $2\times2$ matrix, $$A=\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$$ $\det(A)=ad-bc$ and the matrix, $$B=\begin{pmatrix} 0 & 1 & 0 ...
1
vote
1answer
84 views

Showing a particular map is equivariant with respect to certain group actions

Let $A$ = {triangles in $\mathbb{R^2}$}. We can let $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$ be the vertices of the triangle. The group $GL(2,\mathbb{R})$ acts on $A$ by acting on the vectors of the ...
1
vote
0answers
45 views

How do I know that $\det(a,b)$ is the area of parallelogram?

Please give an easy explanation, high school level.
0
votes
0answers
41 views

How can the area of a parallelogram be show to be equal to the determinant?

I am reading multivariable calculus and it says the area of parallelogram is the same as determinate of 2 by 2 matrix. I googled and found a nice geometric proof showing the base and height to be ...
0
votes
0answers
32 views

Determinant for Area and Volume? [duplicate]

I am studying determinant in Linear Algebra, somehow, I just can't understand determinant is for Area in 2D and Volume in 3D. But how does it make sense to you?