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The image is simply the column space, while the kernel is the nullspace (the set of vectors that map to the 0 vector in the space). It's hard to explain everything without knowing how deep your background in math is.
well i'm in linear algebra and i we're just learning the section on kernel and image and i'm a bit confused while doing the homework. so what i'm gathering is that is you have a matrix with two columns the image of that matrix is v1 and v2. and to find the kernel of the same matrix you would set it equal to zero and solve for the variables?
So, if you have some mapping such that Ax=b, and A were composed of two column vectors v1, v2... The image would be any linear combination of v1 and v2. The kernel would be all solutions to Ax=0.
ok so does the image relate the transformation of x to b?
You can think of the image as the set of all values in the domain that produce a valid mapping to the range. Ax=b is a system of equation characterized by a mapping matrix A. So say A were mxn. A would be mapping some m-dimensional space to another n-dimensional space.
ok so A is a like a function?
If you think about it in terms of mappings, A is like a transformation. Keep in mind that for something to be a function, it has to be one-to-one. If you don't want to think about the heavier math parts, just think of A as the matrix, when applied to some x, produces b (or better yet, the coefficient matrix for a system of linear equations).
ok i got it. so the image is the vectors of A? and the kernel is the solutions when Ax=b=0?
Kernel is when Ax=0.