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Course: Staging content lifeboat > Unit 10
Lesson 4: Matrices- GRAVEYARD Algebra 2: Matrices
- IDEAS Algebra 2: Matrices
- Multiplying a matrix by a column vector
- Transpose of a matrix
- DEPRECATED Matrix transpose
- DEPRECATED Representing relationships with matrices
- DEPRECATED Defined and undefined matrix operations
- DEPRECATED Properties of matrix multiplication
- DEPRECATED Zero and identity matrices
- DEPRECATED Solving matrix equations
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Multiplying a matrix by a column vector
Created by Sal Khan.
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Do vectors in physics have anything to do with vectors in matrices?(14 votes)
Yes. They are different representations of the same data. A vector in physics, what I will call "unit vector notation", may look like the following: 2i+3j. And in "matrix notation": [ 2 3 0 ] (though this is typically column vector, which is difficult to represent here). If I have another vector 4i+5j or [ 4 5 0 ] and we look at some of the operations you will start to notice the similarities.
Unit Vector Notation ¦¦ Matrix Notation
a = 2i+3j ¦¦ a = [ 2 3 0 ]
b = 4i+5j ¦¦ b = [ 4 5 0 ]
a+b = 6i+8j ¦¦ a+b = [ 6 8 0 ]
a-b = -2i-2j ¦¦ a-b = [ -2 -2 0 ]
ab = 23 ¦¦ ab = 23 <= dot product
a x b = -2k ¦¦ a x b = [ 0 0 -2 ] <= cross product
b x a = 2k ¦¦ b x a = [ 0 0 2 ] <= cross product
Where:
i, j, and k are Cartesian unit vectors in the x, y, and z dimensions, respectively.
(It butchered my formatting, the ¦¦ in each row serves as a column divider.)
In the unit vector notation the i term represents the x component, the j term represents the y component, and the k term represents the z component. In matrix notation the first term represents the x component, the second term represents the y component, and the third term represents the z component.
You can see from the examples above that the matrix notation and the unit vector notation yield the same results for the same operations. Either notation is correct, however one notation may be more convenient depending on the tools at hand.(17 votes)
What phenomena or state in real life is represented by this operation? Why was this operation necessary in the first place?(5 votes)
For ex, we need to solve two equations
2x + 3y = 5;
x + y = 2;
I can represent this as product of Matrix A{A is a 2x2 matrix with following rows(2,3) , (1,1)} and column vector w( w is a 2x1 matrix with rows (x),(y)), the resulting matrix is equal to Column Matrix B (B is matrix with rows (5),(1)). By performing certain row and col operations(you'll come to know, if you dont know right now) u can solve for x and y.
You may find using matrix in this scenario is way more complicated than to solve by method of substitution, but matrices reduces the complexity if we have to solve 'n' equations in 'n' dimensions. Hope this helps(3 votes)
Briefly (please), how do vectors differ from n-by-1 matrices? Vectors imply direction in one dimension, but thus far we have only seen examples of vectors displayed as single columns. Are vectors ever displayed as rows instead of columns? I'm not asking about differing notation, but could changing a vector's direction change its application?(3 votes)
I had a similar question a while ago about how vectors differ from matrices and Molly kindly gave me this answer:
"Vectors are used in geometry to simplify three-dimensional problems, and many quantities in physics are vector quantities. A vector has the ability to simultaneously represent magnitude and direction. A matrix, on the other hand, is a rectangular array of numbers which is a key tool in linear algebra. It is used to represent linear transformations and keep track of coefficients in linear equations. Matrices can be made up of row vectors or column vectors. Basically, vectors can be within matrices but not vice versa. Here is a source if you'd like more examples: http://www.differencebetween.net/science/difference-between-vector-and-matrix/"
Hopefully this helps you as it did me!(3 votes)
What is a vector? I don't quite get how he'd multiply two matrices of different dimensions. Please help, am I missing something here?(4 votes)
its like as we multiply A*B so itz A*w n here they are denoting w as vector(1 vote)
I thought w was a matrix...why is it called a vector?(3 votes)
A vector is a 2 x 1 matrix. But as it only has one column, it is 1-dimensional as opposed to the 2-dimensional matrix.(0 votes)
Isn't " w " a 3 x 1 matrix? Why is Sal saying it's a vector?(2 votes)- Because a vector is when you have like one row or column, while a Matrix its an "structure" compound of vectors, but you need more than one to call it a Matrix.(2 votes)
cant we multiply w *A
w is 3*1 and A is 2*3 so the column of w is not equal to the row of a so we shouldnt be able to(2 votes)- In what sense is this really a vector, though? I recall Sal talking about vectors in the Linear Algebra playlist, and I thought that they could be written in either column or row form without changing the meaning. In the case of adding a vector to a matrix, the way it's written (nx1 or 1xn) is very important. So isn't it more correct to say you're smushing a vector into an nx1 matrix, then doing matrix multiplication, rather than multiplying a vector times a matrix?(1 vote)
Think about what a 3-D vector really is. It's just an ordered set of 3 numbers, the x-, y-, and z- components. the vector 3i+4j-5z could just as easily be written as [ 3 4 -5]. It's not squished into a 1x3 matrix, it IS a 1x3 matrix.Turn it on end and it's a 3x1.(3 votes)
I thought vectors were only supposed to be 2X1 matrices are a 1X2 matrix, like points on a graph. How is it that the problem in the beginning is a 3X1?(2 votes)- Does this mean that you cannot multiply same dimension matrices?(1 vote)
- If the matrices aren't square, then yes, you can't multiply a n × m matrix by a n × m matrix.(2 votes)
Video transcript
So we're multiplying
matrix A here by vector w. And they ask us
what is A times w? So let me get my
scratch pad out. And let's think about this. So we have matrix
A times vector w. So just as a reminder,
Aw-- and I'll write it in bold
right over here-- Aw. And I could write it either
as a vector like that, or I could bold that as well. Aw. It's lowercase for a vector,
uppercase for a matrix. This is the same thing
as matrix A times vector w-- let me do that same color--
which is the same thing. This is matrix A. So it's,
A is 0, 3, 5, 5, 5, 2 times vector w. W is the vector 3, 4, 3. So we have a matrix with
two rows and three columns, so it is a 2 by 3
matrix, times a vector. And this is a column vector. It is three rows in one column. So you could view this
as a 3 by 1 matrix. And so, deciding whether this
is even a valid operation, it's just like
multiplying two matrices. You can view this column vector
is really just a 3 by 1 matrix. And the only way that matrix
multiplication is defined is if the columns
in this matrix is equal to the rows
in this matrix. And we see that is
the case, this vector. This matrix, matrix A, has
three columns: 1, 2, 3. And this and vector w
has exactly three rows. So matrix multiplication,
or matrix times vector multiplication,
is defined here. And the way that
we're going to do it, we're going to end
up with a 2 by 1. You could call it
a 2 by 1 matrix. It's going to have two
rows in one column. Or you could view it as
a column vector, when you have two rows
and one column. But either way,
let's actually think about how we would compute it. So it's going to have
two rows and one column. Then I'm going
give a lot of space so that we can actually
do the calculation. So the top entry
right over here, we're going to get
the row information from our first matrix, and
then the column information. There's only one column here. But we're going to get that
from this matrix or this vector, whatever you want to call it. So we're going to have 0 times
3, plus 3 times 4, plus 5 times 3. Now the second entry
right over here is going to be the
second row here. Essentially, the dot product
of that with this vector. And if you don't know
what the dot product is, I'm essentially
about to do that. It's going to be-- you take
each corresponding term, take their product, and then
add up everything together. So you have 5 times 3, plus
5 times 4, plus 2 times 3. And this is going to
simplify to-- This top term, this is a 0. This is 12 plus 15, which is 27. And then the second term,
this is 15 plus 20 plus 6. So this is 35 plus 6 is 41. So it's a column vector,
two rows, 27 and 41. Now let's input that, 27 and 41. So we get 27. And here we can put 41,
and check our answer, and we got it right.