raster - How can I fix a geoTiff with an incorrect CRS?

Trying to load and plot a raster (it's a DEM) in R, but having problems.

Using R I read in a raster file:

library(raster)

dem <- raster("DEM_Victoria.tif")  # >70 MB

print(dem)

class       : RasterLayer 
band        : 1  (of  3  bands)
dimensions  : 5444, 6007, 32702108  (nrow, ncol, ncell)
resolution  : 100, 100  (x, y)
extent      : 176615.6, 777315.6, 5650397, 6194797  (xmin, xmax, ymin, ymax)
coord. ref. : +proj=longlat +ellps=WGS84 +no_defs 
data source : DEM_Victoria.tif
names       : DEM_Victoria

values      : 0, 255  (min, max)

My confusion here is the extent vs the CRS. The extent shows coordinates from a UTM Zone 55S system, while the CRS reports it as longlat.

Similarly, gdalinfo reports:

Driver: GTiff/GeoTIFF
Files:  DEM_Victoria.tif
Size is 6007, 5444
Coordinate System is:
GEOGCS["GDA94",
    DATUM["unknown",

        SPHEROID["unretrievable - using WGS84",6378137,298.257223563]],
    PRIMEM["Greenwich",0],
    UNIT["degree",0.0174532925199433]]
Origin = (176615.587999997427687,6194797.138000000268221)
Pixel Size = (100.000000000000000,-100.000000000000000)
Metadata:
  AREA_OR_POINT=Area
Image Structure Metadata:
  COMPRESSION=LZW
  INTERLEAVE=BAND

Corner Coordinates:
Upper Left  (  176615.588, 6194797.138) (Invalid angle,Invalid angle)
Lower Left  (  176615.588, 5650397.138) (Invalid angle,Invalid angle)
Upper Right (  777315.588, 6194797.138) (Invalid angle,Invalid angle)
Lower Right (  777315.588, 5650397.138) (Invalid angle,Invalid angle)
Center      (  476965.588, 5922597.138) (Invalid angle,Invalid angle)
Band 1 Block=6007x1 Type=Byte, ColorInterp=Red
Band 2 Block=6007x1 Type=Byte, ColorInterp=Green
Band 3 Block=6007x1 Type=Byte, ColorInterp=Blue

Is this real? A problem? (well it is to me, but perhaps because I don't understand something) Fixable?

I tried 'Assign projection' in QGIS, which uses this command:

gdalwarp -t_srs EPSG:28355 -srcnodata -32768  DEM_Victoria.tif DEM_Victoria.tif.tmp

but it failed with:

ERROR 1: Too many points (12100 out of 12100) failed to transform, unable to compute output bounds.

Answer

If I understand this correctly, you're not translating the coordinates, you need to assign the correct CRS. gdal_translate can do this:

gdal_translate -a_srs EPSG:28355 DEM_Victoria.tif DEM_Victoria.tif.tmp