Matrices and linear transformations

[kashiark]

I've recently come to the conclusion that i need to learn matrices. I read that matrices correspond to linear transformations and that every linear transformation can be represented by matrices, but what are linear transformation, and how do you represent it by a matrix?

 

[mathman]

Simplest example (2 dimensions). Let (x,y) represent a point on the plane in some coordinate system. Let (u,v) represent the same point in another coordinate system. Then one can write u=ax+by and v=cx+dy - a linear transformation since x and y appear as first power only. The matrix to represent the transformation will look like:
(a b)
(c d)

The transformation is then U=MX, where M is the matrix, X=(x,y) and U=(u,v). I presume you know how to multiply a matrix times a vector.

 

[kashiark]

i don't understand what is being transformed?

 

[slider142]

Here's another view: Let U and V be vector spaces. Ie., U may be R (real numbers) or R² (ordered pairs of real numbers, commonly represented graphically by the Cartesian plane). A function T from U into V is called a linear transformation if it satisfies the equation T(r*u + w) = r*T(u) + T(w) for all real numbers r and all vectors u and w in U. In other words T distributes over addition and scalar multiplication.
Every n-dimensional vector space has an ordered n-tuple of vectors (v₁, ..., v_n) for which every vector u in that space can be written uniquely as u¹v₁ + ... + uⁿv_n. The numbers uⁱ are called the components of the vector u with respect to the basis (v₁, ..., v_n). For R² for example, a common basis is the two vectors (1, 0) and (0, 1).
Using the component form of a vector and the definition of linear transformation given above, derive the general expression for T(u) where T is a linear transformation from U into V in terms of the numbers uⁱ, the basis vectors e_i of U, and the basis vectors f_j of V.

The following will not make much sense unless you did the above exercise:

You will find that T is specified by certain numbers that show how T maps basis vectors in U into vectors in V, and these numbers naturally have two indices, where the first index corresponds to the basis vector in U and the second to the the component of the transformed basis vector in V.
Matrix multiplication is then defined so that if A is this matrix of numbers corresponding to T with respect to two particular bases, then Ax = T(x).
In particular, if E is a matrix representing the chosen basis of U (in particular, E is just the components of the basis vectors of U written in column form), then note that applying A to E will give you the image of the basis under T. In particular, if E is the standard basis (1s and 0s) then the matrix of T is simply the image of the basis. Ie., if T rotates the axes by 45 degrees (Pi/4 radians), then if our vectors in R² are written with respect to the standard basis ((1, 0), (0, 1)) and we want our image to be written with respect to the standard basis, then the matrix corresponding to T is just the rotated basis ((Sqrt(2)/2, Sqrt(2)/2), (-Sqrt(2)/2, Sqrt(2)/2)). Apply it and check.

 

[HallsofIvy]

I've recently come to the conclusion that i need to learn matrices. I read that matrices correspond to linear transformations and that every linear transformation can be represented by matrices, but what are linear transformation, and how do you represent it by a matrix?

i don't understand what is being transformed?

Then do you understand what a "linear transformation" is? Generally, a "transformation" is a function from one vector space to another. To be a linear transformation, L must satisfy L(u+ v)= L(u)+ L(v) and L(xv)= xL(v) for any vectors a and b and any number x. It is the vector, u, say, that is being transformed to to vector L(u).

One important property of any vector space is that there always exist a "basis" for the space. In particular for a finite dimensional vector space there is a set of vector $\left{ u_1, u_2, \cdot\cdot\cdot, u_n\right}$ such that any vector can be written as a linear combination of those basis vectors: $u= a_1u_1+ a_2u_2+ /cdot/cdot/cdot+ a_nu_n$, for some numbers $a_1, a_2, ..., a_n$, in a unique way.

That means that if we agree on a specific basis, written in a specific order, we can do away with writing the vectors and just write the number: write $u= a_1u_1+ a_2u_2+ /cdot/cdot/cdot+ a_nu_n$ as <a₁, a₂, ..., a_n> and treat it as a vector in Rⁿ.

To write a linear transformation from vector space U to vector space V as a matrix, first select a basis for each, say $\left{ u_1, u_2, \cdot\cdot\cdot, u_n\right}$ for U, and $\left{ v1, v_2, \cdot\cdot\cdot, v_m\right}$ for V. Here, n and m are the dimensions of U and V respectively. Now apply the linear transformation to u₁. The result is in V and so can be written as a linear combination of v₁, v₂, etc. The coefficients for the first column of the matrix representation. Doing the same to u<sub>2[/sup] gives the second column, etc.

Note that choosing a different basis for U, or V, or both will give a different matrix representation of the same linear transformation.</sub>

 

[ZunairaMaryam]

What are Vector spaces,Vector basis,Transformations,linear combination,Linear transformation.I haven't known any of these things?Tell me about these things at a begginer stage.I know generally about simple vectors(not plane vectors)and 3d matrices!
i want to know in simple clear terms,not involving any complex terms!

 

[mathman]

What are Vector spaces,Vector basis,Transformations,linear combination,Linear transformation.I haven't known any of these things?Tell me about these things at a begginer stage.I know generally about simple vectors(not plane vectors)and 3d matrices!
i want to know in simple clear terms,not involving any complex terms!

I advise you to get an elementary text or an elementary course on vector analysis. All those terms would be described there. You might try wikipedia.